Hi, friends I hope you are all well and doing your best in your fields. Today we can discuss the main topic which is acceleration. Acceleration is a central idea in physics that is key to understanding the motion of objects. It is the rate at which an object changes its velocity with time. This variation can take place either as a change in speed or direction. In simpler words, it tells us how fast the velocity of an object is changing along a particular track. As objects move, they may accelerate by gaining, losing, or changing direction. Many concepts in physics and engineering depend on understanding acceleration, such as the movement of celestial bodies to what happens to everyday objects. It is also vital in explaining why objects move, how forces act on them, and the basic principles governing their motion.
In layman’s language, you can experience acceleration when your car speeds up on a highway when an athlete suddenly changes his/her running direction or even if you throw any ball upwards. By studying the acceleration concept, physicists and engineers can predict and explain motion, design efficient transportation systems, and develop technologies based on controlled motion and velocity changes.
Acceleration is simply defined as the rate at which velocity changes in a given period. Velocity is a vector quantity because it has both magnitude and direction; hence, acceleration is also a vector component. The International System of Units (SI) uses meters per second squared (m/s²) as its standard unit of acceleration.
Mathematically, acceleration can be expressed by:
a = Δv/Δt
Where:
∆v( Delta-v ) reflects the change in velocity,
∆t( Delta t ) reflects the change in time.
When an object’s velocity changes uniformly, it undergoes constant acceleration. In contrast, if the rate at which its velocity alters fluctuates, then non-uniform acceleration occurs.
Several types of acceleration can be classified based on their characteristics and context of occurrence:
Uniform acceleration is where the rate at which a body’s velocity changes over time remains the same. Such a kind of motion is common in many theoretical studies, making it an easier method to describe motion.
One instance of uniform acceleration is when an object falls freely in a vacuum under gravity with no other forces acting on it. In such a case, it will accelerate downwards due to gravitational pull at approximately 9.8 m/s².
Non-uniform acceleration refers to when there are varying velocities during different times. This often takes place in real-life instances whereby various forces may be acting on an object at different times. For example, non-uniform acceleration happens to a car that speeds up and slows down in traffic. Alterations in the forces exerted on this vehicle (like gear position or road conditions) make for fluctuations in its rate of change of speed.
In circular motion, there are two components of acceleration: tangential and centripetal. Tangential acceleration occurs whenever there is a change in speed along any point around the circumference. Tangential acceleration is experienced when there is an alteration in the speed along the circular path. On the other hand, centripetal acceleration points towards the center of a circle and helps to keep the object on its curved course. The formula for centripetal acceleration ac can be given as:
ac = v2/r
Where:
( v ) is the velocity of the object along the circular path,
( r ) is the circle’s radius.
Alternatively, in terms of angular velocity (ω) and radius ( r ), we can also write centripetal acceleration:
ac = r ω²
This implies that centripetal acceleration will always point to the center of a circular path and is responsible for altering the direction of an object’s velocity without affecting its magnitude (speed). This concept is important in understanding circular motion because it is related to the force-directed inward that keeps objects following curved paths, which are shown by Newton’s laws of motion.
In physics, tangential acceleration refers to how fast an object’s rate or speed changes as it travels along a curved path. Unlike linear acceleration, which varies an object’s speed on a straight trajectory, tangential acceleration affects the speed of an item moving along a bent direction.
In rotational movement or while an item actions alongside a round route, its pace modifications no longer only in significance but additionally in the course. Tangential acceleration (at) in particular refers back to the factor of acceleration that causes modifications in the pace (importance of speed) of the item. It is directed along the tangent to the route of motion and is measured in meters in line with second squared (m/s²).
At = d|v| /dt
Where |v| denotes the magnitude of the velocity vector v.
Tangential acceleration plays a vital function in expertise round and rotational movement. For instance, whilst an automobile negotiates a curve on a tune, its speed adjustments are not handiest due to speed changes (tangential acceleration) but additionally due to changes in the path (centripetal acceleration). Together, these accelerations determine how easily and predictably an object can navigate curves without skidding or losing manipulation.
In precis, tangential acceleration describes how the speed of an item adjusts along a curved path, offering crucial insights into the dynamics of rotational and round motion in physics.
Centripetal acceleration is the acceleration that continues an item transferring in a circular route. It is directed in the direction of the center of the circle or the axis of rotation, perpendicular to the item’s velocity vector. In many sensible situations, centripetal acceleration arises while an item moves alongside a circular trajectory, together with a car navigating a curve, a planet orbiting a celebrity, or a satellite TV for pc circling the Earth.
Centripetal acceleration is continually directed toward the center of the round course and is chargeable for converting the route of an item’s pace without altering its velocity (value of speed). It is a vital idea in information circular movement and is carefully related to the centripetal pressure required to preserve the object’s round path, as described with the aid of Newton’s laws of motion.
In summary, centripetal acceleration is the acceleration that acts toward the center of a round direction, making sure that gadgets preserve their trajectory and do now not deviate into an instant line.
The mathematical representation of acceleration can be explored through various equations and principles, often linked to Newton’s laws of motion.
For motion with uniform (constant) acceleration, we use a set of kinematic equations to describe the relationships between displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t). These equations are fundamental in classical mechanics and are used to solve various problems involving motion.
v = u + at
According to this equation, the final velocity (v) is equal to the initial velocity (u) plus the product of acceleration (a) and time (t).
s = ut + ½ at²
This equation provides the displacement (s) of an object based on its initial velocity (u), time (t), and acceleration (a).
v² = u² + 2as
This equation relates the final velocity (v), initial velocity (u), acceleration (a), and displacement (s). It is especially helpful in situations when time is unknown.
These kinematic equations assume constant acceleration and are widely used in physics to analyze the motion of objects.
Classical mechanics is based on Newton’s second law of motion. Establishes the relationship between force (F), mass (m), and acceleration (a);
F = ma
This equation shows that the force acting on an object is equal to the product of its mass and velocity. It means that the velocity of an object is inversely proportional to its mass and directly proportional to the applied force. This principle is important for understanding how forces affect processes.
For example, if a net force is applied to a car, its acceleration can be determined by dividing the force by the weight of the car. Conversely, the required force can be calculated by determining the desired magnitude and velocity.
When an object acts in a circular direction, it experiences centripetal acceleration, which is directed toward the center of the circle. This acceleration is essential to trade the direction of the item's speed, keeping it on the circular course. The following yields the centripetal acceleration (ac ) ;
ac = v²/r
where:
(v) is the velocity of the object along the circular path,
(r) is the circle’s radius.
This equation suggests that centripetal acceleration increases with the square of the velocity and reduces with an increase inside the radius of the circular course. It is crucial for understanding the dynamics of items in circular motion, like cars taking a turn or planets orbiting the Sun.
Example:
Suppose a car traveling at a speed of 20 m/s around a curve with a radius of 50 meters. The centripetal acceleration ( ac ) Can be calculated using the formula:
ac = v²/ r
ac = 20²/ 50
ac = 400/50
ac = 8m/s²
The car exploits a centripetal acceleration of 8 m/s² in the direction of the circular path’s center.
In calculus, acceleration can be expressed as the derivative (d) of velocity concerning time. If velocity (v) is a function of time (t), then acceleration (a) is given by:
a = d v / dt
This differential form provides a precise way to describe acceleration at any given instant. For cases where velocity is not constant, integrating acceleration over time can determine the change in velocity;
v = ∫a dt
Similarly, if acceleration itself is a function of time, position, or velocity, the motion of the object can be analyzed using differential equations. Solving these equations yields detailed information about the object's velocity and position over time.
In particular, the uniform acceleration with which the bodies falling freely near the Earth’s surface act is called gravitational acceleration or Gravitational acceleration is the acceleration associated with the force of gravity of a body, for example, the Earth. Estimates put it at 9. 8 m/s² and this is abbreviated as ‘g’. The equations of motions for objects under gravitational acceleration are as follows; very much like kinematical equations, the only changes that occur is the use of (a) as (g).
For an object in free fall, the equations become:
Final velocity: v = u + gt
Displacement: s = ut + ½ gt²
Velocity-displacement: v² = u² + 2gs
These equations are very useful for solving problems on the motion of objects under the force of gravity like a falling object or a projectile.
Gravity acceleration is simply the acceleration that an object goes through when under the effect of gravity. On the surface of the earth, the acceleration due to gravity is roughly about 9. 81 meters per second squared ( 9. 81 m/s² ). This value gives the rate at which objects fall towards the center of the earth because of the force of gravity.
Of the two types of acceleration, the tangential acceleration deals with acceleration in the plane of motion or along the circle’s circumference while radial acceleration refers to the acceleration that is directed toward or away from the center of the circle. Tangential acceleration (at) is an acceleration in the radial direction it is incorporated into the change speed of the object in the circular path and radial or centripetal acceleration (ar) is towards the center to reverse the direction of the velocity vector head.
If the angular velocity (ω) of an object in circular motion changes over time, the tangential acceleration can be expressed as
If the angular velocity (ω) of an object in circular motion changes over time, the tangential acceleration can be expressed as:
at = r dω/ dt
where (r) is the radius of the circular path.
The total acceleration (a) is the vector sum of tangential and radial acceleration which is represented by The mathematical formula for the same is;
a = √at² + ar²
These components are significant in analyzing objects in a non-uniform circular motion because it is applicable in real-life situations like cars moving at different speeds on curves.
Concluding, it is necessary to state that acceleration in mathematics is described by several equations and principles that determine the change of velocity in an object throughout time. Some of them are the kinematics equation for the impact of uniform acceleration, Newton’s second law, centripetal acceleration for circular motion, differential calculus for non-uniform acceleration, specific instances like gravitational acceleration, and so on. For students, these tools are indispensable necessities for finding solutions to various problems and for understanding the motion of various objects in situations of their application.
Accelerometers are therefore instruments that are used in the measurement of acceleration. It is used in almost every technological field; right from mobile phones to space shuttles. Accelerometers work based on the principle that they generate an electrical signal that is proportional to the force exerted on a test mass in the device under the effect of acceleration. Below are some of the common types of accelerometers;
These measure changes in capacitance due to the displacement of a test mass when acceleration happens. capacitive accelerometer is a kind of accelerometer which its
working is based on capacitance change as a result of acceleration.
Capacitive accelerometers refer to the devices that are commonly used in the measurement of acceleration through capacitance variations. They consist of two main components: a proof mass and a fixed capacitor structure is used. The proof mass is often, but not necessarily, small and is free to move in a direction perpendicular to the direction of the force being measured by the interferometer, and is attached between two springs. As the accelerometer goes through acceleration, the proof mass shifts concerning the fixed plates of the capacitor and results to a change in capacitance.
Based on the above results, it can be noted that the change in capacitance is directly proportional to the acceleration applied to the device. This can be measured electronically and converted into an acceleration value depending on calibration value and signal processing measures.
The above capacitive accelerometers' main characteristics and benefits are high sensitivity, low power consumption as well as an extended measurement range. They are popular in many demands for the measurement of acceleration including automotive technology (For example air-bag systems), aerospace engineering (For instance, flight control), handheld devices (For instance, the built-in accelerometer in every smartphone for tilting), and large machinery (For instance vibration measuring).
These produce an electric charge as a result of mechanical stress from acceleration. A piezoelectric accelerometer, as a kind of accelerometer, is an instrument that can define acceleration by using the piezoelectric phenomenon.
Piezoelectric accelerometers are devices used to measure accelerations, vibrations, and shocks for many applications. They use the working of piezoelectric effect which states that acceleration is transformed into electrical charge. These accelerometers include a piezoelectric crystal, usually of quartz or ceramic, whose output voltage depends on the acceleration.
The important characteristics of piezoelectric accelerometers are high or greater sensitivity, wide frequency range, robust structure, and insensitivity to high temperatures. These applications are in automobile manufacturing, aerospace, manufacturing industries, and in the monitoring of structures.
Piezoelectric Crystal: The principal part of a piezoelectric accelerometer is a piezoelectric crystal with quartz or ceramic material. When the accelerometer does move and thus, apply inertial force on it, then, the crystal elongates or contracts slightly.
Generation of Electrical Charge: As per the piezoelectric effect, the above-mentioned deformation results in the formation of an electrical charge on the surface of the crystal. The degree of charge is proportional to the acceleration applied to the accelerometer device.
Measurement and Output: The generated electrical charge is usually intensified and analyzed through the use of electronics within the accelerometer. This leads to an output signal (generally voltage) proportional to the acceleration that the device is subjected to.
Piezoelectric accelerometers are provided in several formats and orientations single, dual, and triaxial depending on the utilization of the device. They are widely applied for vibration measurement in rotating and other machinery, structure monitoring, automotive safety by crash testing, and seismic application.
MEMS accelerometers are small-size, low-cost acceleration sensors commonly used in portable electronics. They employ minuscule mechanical elements that bend with acceleration and change the various electrical currents. MEMS accelerometers are electrical devices that are quite small and are employed for measuring accelerations.
MEMS accelerometers as small, integrated devices that measure acceleration using microfabrication technology. They can be classified into the family of micro-electromechanical system sensors, often abbreviated as MEMS, where mechanical components include springs and proof masses as well as electronic components like capacitors and resistors and are all integrated into one chip.
Microfabricated Structure: Generally the MEMS accelerometers have a micro-fabricated structure; in most common configurations there is a proof mass that is attached to the substrate by springs. The proof mass is displaced relative to the package frame under the action of external forces that cause acceleration.
Sensing Principle: Acceleration leads to forces on the proof mass causing it to move a certain distance away from the springs. This displacement is measured with several sensors, for example, capacitive sensors or piezoresistive sensors.
Signal Processing: Due to this movement of the proof mass an electrical signal corresponding to the acceleration is obtained. The signal is then amplified by integrated electronics which are provided in the specific MEMS chip to generate a readable output.
There are different types of MEMS accelerometers and they can be classified based on their axis; they can have single-axis, two-axes, or three-axes. They are used in a wide range of applications, including:
Consumer Electronics: Smartphones and tablets motion sensing, and gaming control equipment.
Automotive: Airbag: active and passive electronic systems concerning vehicle stability, and vehicle navigation systems.
Industrial: Machinery condition monitoring, attitude determination and control, and structure health monitoring.
Healthcare: Including fall detection gadgets, wearable health monitors, and health enhancers in the human body.
In the realm of special relativity, the concept of acceleration becomes more complex. As objects approach the speed of light, their mass effectively increases, requiring greater force to achieve further acceleration. The relativistic form of Newton's second law is used to account for these effects:
F = γ³ma
where:
γ is the Lorentz factor, given by ;
v is the velocity of the object,
c is the speed of light.
This equation shows that as v approaches c, γ increases significantly, making it increasingly difficult to accelerate the object further.
In general relativity, acceleration due to gravity is understood as the curvature of space-time caused by massive objects. Instead of viewing gravity as a force, general relativity describes it as the effect of space-time curvature on the motion of objects. The equations governing this curvature are given by Einstein's field equations:
Gμν = 8πG/c⁴ Tμν
where:
Gμν is the Einstein tensor representing space-time curvature,
G is the gravitational constant,
c is the speed of light,
Tμν is the stress-energy tensor representing matter and energy distribution.
In this framework, objects follow geodesics, which are the paths determined by the curvature of space-time. The acceleration of objects in a gravitational field is thus a manifestation of their motion along these curved paths.
In aerospace engineering, understanding acceleration is crucial for designing aircraft and spacecraft. Engineers must consider the accelerations experienced during various phases of flight, including take-off, cruising, maneuvering, and landing.
During the design of aircraft, engineers analyze the accelerations and forces acting on the airframe to ensure structural integrity and passenger comfort. This involves studying the effects of aerodynamic forces, thrust, and gravitational forces. Pilots also need to understand acceleration to perform safe and efficient maneuvers.
Spacecraft navigation relies heavily on precise calculations of acceleration. For example, during interplanetary missions, spacecraft use gravity assists (also known as gravitational slingshots) to change their trajectories and speeds. By carefully planning these maneuvers, mission planners can use the gravitational fields of planets to accelerate the spacecraft, saving fuel and reducing travel time.
In the automotive industry, acceleration is a key factor in vehicle performance, safety, and comfort. Engineers use the principles of acceleration to design better engines, braking systems, and suspension systems.
The performance of a car engine is often measured by its ability to accelerate the vehicle. Engineers design engines to provide optimal acceleration while maintaining fuel efficiency and reducing emissions. This involves optimizing the combustion process, transmission system, and overall vehicle weight.
Modern cars are equipped with advanced safety systems that rely on precise measurements of acceleration. For example, electronic stability control (ESC) systems use accelerometers to detect and correct skids or loss of control. Similarly, airbag deployment systems use accelerometers to detect sudden decelerations during collisions and deploy airbags to protect occupants.
In robotics, controlling acceleration is essential for achieving precise and efficient movements. Robots in manufacturing, healthcare, and exploration require accurate acceleration control to perform tasks effectively.
Industrial robots often operate in environments that require high precision and speed. To achieve this, robots use advanced control algorithms to manage acceleration and deceleration smoothly. This helps prevent mechanical wear and tear and ensures accurate positioning.
Medical robots, such as surgical robots, must operate with high precision to perform delicate procedures. Controlling acceleration in these robots is critical to avoid damaging tissues and to provide smooth and controlled movements.
Various experimental techniques are used to measure and analyze acceleration in different contexts. These techniques range from simple mechanical setups to advanced electronic systems.
One of the simplest ways to study acceleration is through pendulum experiments. By measuring the period of a pendulum's swing, one can calculate the acceleration due to gravity. For the simple pendulum, The period (T) is written as:
T = 2π √L/g
where:
( L ) is the pendulum’s length,
and ( g ) is the acceleration caused by gravity.
By measuring (T) and (L), the value of (g) can be determined.
Drop tower experiments involve dropping an object from a known height and measuring its acceleration as it falls. These experiments are often used to study the effects of microgravity and to test the performance of accelerometers. The object's acceleration can be analyzed using high-speed cameras or precise timing systems.
Computational methods, including numerical simulations and modeling, are used to study complex systems where analytical solutions are not feasible.
Finite element analysis (FEA) is a computational technique used to simulate and analyze the behavior of structures under various loads, including acceleration. FEA divides a structure into smaller elements and solves the equations of motion for each element. This method is widely used in engineering to predict the performance of complex systems.
In fluid dynamics, acceleration plays a critical role in the behavior of fluids. Computational fluid dynamics (CFD) is used to simulate the motion of fluids and the forces acting on them. By solving the Navier-Stokes equations, CFD can provide detailed insights into the acceleration of fluid particles and the resulting flow patterns.
Acceleration is a vital idea in physics that narrates how the object’s velocity varies over time. It is essential for understanding the dynamics of motion and has a wide range of applications in everyday life, engineering, technology, space exploration, and scientific research.
From the uniform acceleration of free-falling objects to the complex accelerations experienced by spacecraft, the study of acceleration provides insights into the forces and interactions that shape our world. By employing mathematical equations, experimental techniques, and computational methods, scientists and engineers can analyze and predict the behavior of systems under various conditions of acceleration. Understanding acceleration not only helps us comprehend the physical world but also drives technological advancements and innovations that improve our lives.
Hello friends, I hope you are all well and doing good in your fields. In the previous article, we can discuss the distance and displacement of the objects. Still, today we can talk about velocity because velocity provides information about the speed from which the object can be displaced from one point to the other point. Everything can be moved from one place to another with different speeds and velocities. To understand the speed or the motion of an object it is compulsory to understand the velocity. The concept of velocity is the cornerstone notion in the field of physics because it can provide information about the rate of change of displacement or position. Through speed and velocity, the object can be displaced but the speed is a scalar quantity because it can only provide the magnitude but the velocity is the vector quantity which can provide both magnitude and direction.
In dynamics, and kinematics it can play a very crucial role in understanding the motion of the displaced objects and also helps to understand the various physical phenomena related to the motion of an object. Simply speed and velocity are a combination but speed only provides the magnitude while velocity provides both the magnitude and direction of the moving object. Like kinematics, the notion of velocity is fundamental to understanding the motion of an object in mechanics.
Both speed and velocity describe the motion of an object, based on the idea that an object can be moved fast or slow. Speed and velocity help to identify the objects that can be moved fast or also those objects that can be moved slow. When two objects are displaced at the same time, the fastest object can reach its ending point and this object has high speed and velocity as compared to others. Now, we can start exploring the basic definition of velocity, its types, mathematical representation, examples, and the main difference between speed and velocity.
Velocity is defined as:
“The rate of the change of the position of the body with time or the rate of the change of displacement with time is termed as velocity.”
The main difference between speed and velocity is that speed is the rate of change of position in the unit of time but velocity is the rate of change of displacement in the unit of time.
Mathematical representations of velocity are written below:
Velocity = displacement x time
v =dt
there,
v represented velocity and d represented displacement and t represented the time.
This formula can also be written as:
d = vt
This formula shows that the body can be displaced with some velocity at the unit time.
The SI unit of the velocity is the same as the speed which is meter per second and is written as ms (ms-1).
Dimension of velocity SI unit are written below:
ms-1 = LT-1
There, L is the dimension for m and T-1 is dimension or s-1.
Average velocity can be defined as:
“ the ratio of the total change of position or displacement with the total time taken is termed as average velocity”
generally, the average velocity is the overall motion of the object that can be covered from one place to another with a unit of time.
Average velocity can be mathematically represented as:
vav = ΔdΔt
There, v represented the average velocity, Δd represented the rate of change in displacement and Δt represented the rate of change in time.
The SI unit of the average velocity is a meter per second and is written as ms (ms-1)
Dimension of average velocity SI unit are written below:
ms-1 = LT-1
There, L is the dimension for m and T-1 is dimension or s-1.
The direction of the average velocity of the displaced object is always in the direction of the displacement.
Some major limitations of the average velocity are given there:
Average velocity can't describe the motion of the object or body, the body or object may be do random motion or maybe it can do a steady motion. But the average can't tell about their type of motion.
Average velocity can't also provide information about the path in which the body can be displaced because the path may be curved or it may be straight.
Instantaneous velocity can be defined as:
“The instantaneous velocity of the object is the limit or average velocity interval with a change in the time and the velocity reached at zero.”
the instantaneous velocity is the velocity of the moving object at some special or specific point or moment. Instantaneous velocity can be derived when we take the average velocity at a specific time.
The mathematical representation of the instantaneous velocity is written below:
vins = Δt0ΔdΔt
There, vins represented the instantaneous velocity, Δt0represented the average limit of the velocity with time, and reached zero, Δd the rate of change of displacement, and Δt represented the rate of change in time.
The formula of the instantaneous velocity can also be written as:
s = v dt
There, s represented the displacement function, v represented the velocity, d represented the displacement of the displaced object and t represented the time in which the body can be displaced.
The SI unit of the instantaneous velocity is the same as the velocity or the average velocity which is a meter per second and is written as ms (ms-1)
Dimension of instantaneous velocity SI unit is written below:
ms-1 = LT-1
There, L is the dimension for m and T-1 is dimension or s-1.
The direction of the instantaneous velocity of the displaced object is always in the direction of the displacement.
The instantaneous velocity of the moving body can't equal zero but the average velocity of the moving body or the object may be equal to zero.
Uniform velocity can be defined as:
“The body that can cover the displacement is equal to the time intervals and moves constantly without changing the displacement in unit time.”
The body can be moved with a uniform velocity when it can cover the displacement, the displacement is equal to the time interval and they are constantly moved.
Non-uniform velocity can be defined as:
"The body that can cover the unequal displacement is equal to the time intervals and moves constantly with changing the displacement in unit time."
Some other units of velocity rather than meters per second are mph or fts-1These two units are also commonly used.
Many different particles can be moved or displaced at the same time with uniform velocity in many different special cases but in these cases, the particles moved with uniform velocity with different time intervals such as v1t1, v2t2, ………. vntn. then we can find the average velocity by using the given formula.
v = v1t1 + v2t2 + v3t3 +v4t4........... + vntnt1 + t2 + t3 +t4........ + tn
Then, we know that t1+ t2 + t3+ …… + tn = t
Now we use the arithmetic mean for the average speed and write as:
v =v1 + v2+ v3 +v4........... + vnn
Also written as:
v = 1n i=1nvi
But when the particles can be displaced with different or numerous distances with equal intervals of time and also with the same distance then it can be written as:
v = s1+ s2 + s3 + s4...... snt1 + t2 + t3 +t4........ + tn
It can also be written as:
v = s1+ s2 + s3 + s4...... sns1v1 + s2v2 + s3v3 + s4v4+ ......+ snvn
In the average speed, we can use the arithmetic mean but when the particles cover different distances then it can be equal to the harmonic mean and it can be written as:
v = n (1v1 + 1v2 + 1v3 + 1v4 + ……. + 1vn)-1
And then according to harmonic mean, it can be written as:
v = n (i=1n1vi)-1
Some different quantities which can be depended upon the velocity are given there:
Drag force
Momentum
Escape velocity
Kinetic energy
Lorentz factor
Their description is given there:
Drag forces are the fundamental concept to understand for understanding the motion of fluid or in fluid dynamics because drag force is the specific force that can act opposite to the object's motion because the body or object can be moved in the fluid so this force helps to move in the fluid in the right direction. The formula for drag force is vine there:
FD = 12 ρ v2 CDA
According to the given formula, it can be shown that the drag force is dependent upon the square of the velocity.
There,
FD represented the drag force in fluid dynamics.
Ρ represented the momentum of the moving object.
v2 is the square of the velocity.
CD is the coefficient of the drag force.
A represents the area in which the drag force or the body can be moved.
The momentum of the body can also depend upon the velocity directly because according to the second law of Newton, the momentum of the body or object is equal to the product of mass and velocity and it can be written as:
ρ = mv
There,
ρ represented the momentum
M represents the mass of the object or body
v represented the velocity of the object through which it can be moved or displaced.
Simply, escape velocity is the velocity of an object to escape from the earth moon or other massive bodies. Through the escape velocity, the rockets can reach space. The general formula that can describe the velocity is given there:
ve = 2GMr
There,
ve represented the escape velocity.
G represented that gravitational force and its value are fixed.
M represented the mass of the object or body
r represented the radius of the body from Earth.
The above formula can also be written as:
ve = 2gr
Kinetic energy can also depend upon the velocity directly and the formula that can show this is given there:
Ek = 12 mv2
There,
Ek represented the kinetic energy.
m represented the mass of the object or body
v represented the velocity and kinetic energy was directly dependent upon the square of the velocity.
In the formula of special relativity of the Lorentz factor, the Lorentz factor depends upon the velocity, and the formula is written there:
🇾 = 11- v2c2
There,
γ represented the Lorentz factor.
V represented the velocity on which the Lorentz factor depends.
c is the speed of light.
When the velocity of the object or the moving body occurs in one dimension, then their velocity is always scalar and it can be written as:
v = v - (-w)
When the two objects can be moved in the same direction the equation can be written like this. But if the two objects are not moved or displaced in the same direction and move in opposite directions then it can be written as:
v = v - (+w)
In the equation of motion velocity is the main integral because through the velocity we can understand the motion of the numerous displaced objects when different forces act upon that. The equation of motions in which the velocity relationship is shown is written below in detail.
The first equation of motion
The second equation of motion
Third equation of motion
The formula of the first equation of motion in which the velocity relation can be shown is given there:
v = vi + at
There,
v is the final velocity, vi represents the initial velocity, a represents the acceleration and t represents the time.
The second equation of motion in which the velocity relationship with the equation is shown is written there.
s = xit + 12 at2
There, s represented the displacement of the moving object, xi represented the initial velocity, a represented the acceleration and t represented the time.
The third equation of motion can be used to analyze the objects that can fall from the height or can also be used to analyze those objects or bodies that can be moved on the highway with high velocity or speed.
v2 = xi2 + 2as
There, v represented the velocity, xi represented the initial velocity, a represented the acceleration of the moving object and s represented the displacement.
Somehow the velocity and the speed are the same but their nature is different because speed is the scalar quantity and the velocity is the vector quantity because it can provide both magnitude and direction. the direction of the velocity is along with the displacement and their direction shows the final point or destination of the displaced object while the magnitude of the velocity is always their speed. Due to the vector nature of the velocity we can use different vector algebraic methods to add or subtract the multiples or the complex velocities efficiently. Numerous objects have different velocities that can act on the body at the same time so understanding them is crucial to understanding the fundamental concept of velocity.
The major differences between the speed and the velocity are given there:
Speed |
Velocity |
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Velocity can be calculated easily but its calculation depends upon the dimension in which the velocity occurs. It can also depend upon the cartesian coordinate system which can be used to represent the velocity. if the velocity can lie on one dimension then it can be calculated easily by dividing the total displacement by the total time taken which can be used by the object to change its position. but if the velocity can lie on the cartesian plane with two or three dimensions then to solve their complexity we can use different vector rules for their division. Some calculations of velocity according to dimension with examples are given there:
When the body can be moved in one dimension then the value of velocity may be positive or negative because the negative and positive velocity describe the direction of the moving body in a single dimension on the x-axis. When the velocity can occur in one dimension then for their calculation we can use the given formula:
v =x2- x1t
There, v represented the velocity and x1 , x2 represented the change of the body from the initial point to the final point and t represented the time taken that can be used to change the position. As we know,
△d = x2 – x1
So the formula can slo be written as,
v = Δdt
As we know Δd represents the displacement of an object.
The complex velocity can occur in more than one dimension like in a dynamic field when two or more objects can be moved then it is complex so we can use the vector rules and it can also help to understand the motion of the body crucially. If the body can be moved in two dimensions, it starts its velocity from the point ( x1, y1 ) to the endpoint ( x2, y2 ) in some time. Then to calculate the velocities we can use the given formula which can be written below:
v = (x2- x1) i + (y2 - y1) jt
There, i, j are the unit vectors of the x and y directions in the cartesian coordinate system.
The complex velocity can occur in more than one dimension like in a dynamic field when two or more objects can be moved then it is complex so we can use the vector rules and it can also help to understand the motion of the body crucially. If the body can be moved in three dimensions, then it starts its velocity from the point ( x1, y1, z1 ) to the endpoint ( x2, y2, z2 ) in some time. Then to calculate the velocities we can use the given formula which can be written below:
v = (x2- x1) i + (y2 - y1) j + ( z2- z1) kt
There,i, j and k are the unit vectors of the x and y directions in the cartesian coordinate system.
Example of velocity in one dimension:
Suppose the body or an object can be moved from the initial point with the speed of 3m and reach the end with the speed of 7m in 5s. Then find the velocity when the body is displaced in the right or positive direction.
To find:
v =?
Given:
x1 = 3m
x2 = 7m
t = 5s
Formula:
v =x2- x1t
Solution:
By using the above formula
v =x2- x1t
By putting the values
v = 7 - 35
v = 45
v = +0.8 ms-1
But if it can move to reverse and move negative direction then,
x1 = 7m
x2 = 3m
t = 5s
By using the formula
v =x2- x1t
By putting the values
v = 3 - 75
v = -45
v = – 0.8 ms-1
Example of velocity in two dimensions:
suppose the body or an object can be moved from the initial point with the speed of (3m, 4m) and reach the end with the speed of (7m, 8m) in 4s. Then find the velocity, magnitude, and direction.
To find:
v =?
v = ?
θ = ?
Given:
x1, y1 = 3m, 4m
x2, y2 = 7m, 8m
t = 4s
Formula:
v = (x2- x1) i + (y2 - y1) jt
Solution:
By using the above formula
v = (x2- x1) i + (y2 - y1) jt
Now by putting the values
v = (7- 3) i + (8 - 4) j4
v = 5+44 , v = 1.25 i + 1j ms-1
Magnitude:
Now apply the formula of magnitude
v = (A)2+ (B)2
Putting the values in the formula
v = (1.25)2+ (1)2
v = 1.7 + 1
v = 1.6 ms-1
Direction:
To find the direction we can use the given below formula:
θ = tan-1(BA)
Now by putting the values
θ = tan-1(11.25)
example of velocity in three dimensions:
suppose the body or an object can be moved from the initial point with the speed of (3m, 4m, 6m) and reach the end with the speed of (7m, 8m, 10m) in 4s. Then find the velocity and magnitude.
To find:
v =?
v = ?
Given:
x1, y1, z1 = 3m, 4m, 6m
x2, y2, z2 = 7m, 8m, 10m
t = 4s
Formula:
v = (x2- x1) i + (y2 - y1) j +( z2- z1) k t
Solution:
By using the above formula
v = (x2- x1) i + (y2 - y1) j +( z2- z1) k t
Now by putting the values
v = (7- 3) i + (8 - 4) j + (10-6) k4
v = 5+44 , v = 1.25 i + 1j + 4k ms-1
Magnitude:
Now apply the formula of magnitude
v = (A)2+ (B)2 +(C)2
Putting the values in the formula
v = (1.25)2+ (1)2+ (4)2
v = 1.7 + 1+ 16
v = 4.32 ms-1
Velocity is the vector quantity or sometimes it becomes more complex so it can be represented in graphic form. The graphic representation of velocity is given there:
Velocity time graph
Position time graph
In the velocity time graph, the whole area which is under the graph represents the displacement that can be covered by the moving object or body or the slope in the velocity time graph represents the acceleration of the moving object.
In the position-time graph of the velocity the straight curve represents that the velocities are constant, curved slope represents that the velocities are not constant but if the slope suddenly becomes curved then it indicates that the velocities are instantaneous. and all of these velocities with their graph are drawn below.
Velocity can play a very fundamental role in different fields like physics, dynamics, and many others some of their applications are given there:
Medicine: in the field of medical sciences, the concept of velocity is used to understand the blood flow or the movement of body parts.
Navigation: velocity helps the pilots and sailors control the speed of the boat and ship.
Engineering: design the new machines after analyzing the velocity.
Sports: velocity helps athlete increase their performance in swimming, running, or other sports activities.
Mostly the complex problems of velocity can be solved by using the given system because they are vectors and vectors are complex and solved by the algebraic methods in some coordinate systems. the higher dimensions problems can be solved in the given coordinate systems.
Cartesian coordinate system
Spherical coordinate system
Polar coordinate system.
In the field of physics, concept of the velocity is crucial to understand because it helps to understand the motion of a moving object efficiently. Velocity can provide the quantity measurement of the moving object, due to its vector nature it can provide both magnitude and direction also.in dynamics or kinematics, velocity helps to understand the behavior of the moving body or object because it can play a very fundamental role in the motion of all bodies or objects. After reading this article the reader can find or understand the velocities that can occur in one direction or dimension or may occur in more than two or three dimensions.
Hi friends, I hope you are well and doing good in your fields. Today we can discuss the main topic of displacement which plays a fundamental role in the motion of the object. In simple words to study and understand the motion of objects, the concept of displacement is fundamental. When the object can moved it can change its position and cover some distance and displacement. Both quantities represented the quantitative information or description of the motion of an object. Sometimes both quantities of distance and displacement are understood the same but they are not similar to each other because in distance we can describe the motion of an object as the object can be moved from one position to another position but in displacement we can measure the distance with time.
To understand the displacement it is compulsory to also understand the concept of the position of the object and the distance because after understanding both all these quantities we can efficiently understand the motion of the object in quantitative form.in different fields of physics like dynamics and kinematics displacement concepts are fundamental. Now in this article, we can also discuss the difference between distance and displacement but mainly we can explore the definition of displacement, its mathematical expression, significance, and examples.
Displacement provides a quantitative description of the distance of the object that can covered by the object from its initial position to the final position. Simply we can define displacement as:
"The object that can be moved and change its position from its initial position to the final position is known as displacement."
Displacement provides information about the direction of the object and also provides information on how far the object can be displaced.
Displacement is a vector quantity so that's why it can provide both magnitude and the direction of the object which are in motion.
The displacement can be represented through an arrow or also in bold letters because they are the vector quantity. For example d or as d.
Mathematical expression and formula of displacement are given there:
d = rf - ri
It can also be written as:
d = B - A
B and A represented the initial and the final position.
Or also as;
Δx = xf - xi
Their Δx represented the change in displacement and xf and xi represented the final and initial position of the moving object.
There,
d = represented the displacement in which the object can change the position from the initial to the final position.
rf= represented the final position in which the object can be displaced.
ri = represented the initial position in which the object can be displaced.
The SI unit of the displacement is the meter. And it can be represented as m.
The magnitude of the displaced object is always equal to the length of the line in which the object covers the distance from the initial position to the final position.
Some examples of displacement are given there:
Let us consider the teacher who can take the lecture and use the whiteboard the initial point is when the teacher starts writing and the final point is when the teacher stops writing then the initial point value and the final point value can be subtracted and provide the displacement.
The concept of the position of the object is compulsory to understand because it is the base or fundamental concept in the motion of the object. Even distance and displacement can be described after understanding or knowing the position of the object accurately. In the field of physics, we can discuss motion in kinematics, dynamics, and many others so it is a must to clear the concept of position.
In simple words, the position of the object is the place or area in which the object can moved or placed. In the motion of an object when the object starts its distance or work then it is referred to as the initial point or position or when the object stops moving or stops working then it can be referred to as the final point or position.
For instance, the teacher can take the lecture and use the whiteboard the initial point or position is when the teacher starts writing and the final point or position is when the teacher stops writing then the initial point value and the final point value can be subtracted and provide the displacement. Like this, every object that can be moved has some starting position where it can be placed and some final point where it stops. everything has different positions in which they are placed but in physics to understand motion, displacement, distance, and velocity it is fundamental to clear the concept of position also.
After understanding the position of the object now we discuss the main topic of distance, many people are confused or assume that both distance and displacement are the same but they are not the same they are different from each other. In simple words, distance are movement of the moving object from one point to another point and it can describe the direction also but displacement provides the both direction and magnitude of the moving object.
In the given figure the distance and displacement are both shown. The body that can start its journey from its starting point and end its some final point the whole path that can be covered by the body or the longest path that can be covered by the body or moving object is termed as the distance but the displacement is the shortest distance which can be covered by the moving body or an object.
The major differences between the distance and the displacement are given there:
Distance |
Displacement |
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Example no 1:
Consider the person who can travel a distance of 150 m to the south and travel to the north about 205 m to do some work. Now calculate the displacement that can be covered by the person.
To find:
Displacement = Δx = d = ?
Formula:
The formula that can be used to find the displacement is:
Δx = xf - xi
Given:
xi= 0
xf = 150 m – 205 m
Solution:
Δx = xf - xi
By putting the values of the final and initial position of the object
Δx = (150 m – 205 m) - 0
The four different objects can be moved and have different initial points and final points and they are opposite to each other as shown in the given figure below there:
Now calculate the displacement of these four objects A, B, C, and D which can have different initial and final points.
To find:
Displacement = ΔxA = dA = ?
= ΔxB = dB = ?
= ΔxC = dC = ?
= ΔxD = dD = ?
Displacement for the object A:
Given:
According to the given figure, the initial position value and the final position value for the object A are given there:
xi = 0 m
xf = 7 m
Solution:
By using the displacement formula which is given there,
ΔxA = xf - xi
Now putting the values
ΔxA = 7 m - 0 m
ΔxA = + 7 m
Displacement for the object B:
Given:
According to the given figure, the initial position value and the final position value for the object B are given there:
xi = 12 m
xf = 7 m
Solution:
By using the displacement formula which is given there,
ΔxB = xf - xi
Now putting the values
ΔxB = 7 m - 12 m
ΔxB = - 5 m
Displacement for the object C:
Given:
According to the given figure, the initial position value and the final position value for the object C are given there:
xi = 2 m
xf = 10 m
Solution:
By using the displacement formula which is given there,
ΔxC = xf - xi
Now putting the values
ΔxC = 10 m - 2 m
ΔxC = + 8 m
Displacement for the object D:
Given:
According to the given figure, the initial position value and the final position value for the object D are given there:
xi = 9 m
xf = 5 m
Solution:
By using the displacement formula which is given there,
ΔxD = xf - xi
Now putting the values
ΔxD = 5 m - 9 m
ΔxD = - 4 m
The four objects that are moved in the tiles are the different initial and final point values. The tiles' sides square are equal to 0.7 km. By understanding the given figure calculate the distance that can covered by all four objects and also calculate the magnitude of displacement of all 5 different objects.
To find:
The total distance that can be covered by the 5 objects = D =?
Given:
AB = 4.2
BC = 1.4
CD = 2.1
DE = 0.7
EF = 0.7
Formula:
The formula which can be used to calculate the total distance is given there
D = AB + BC + CD + DE + EF
Solution:
By using the formula
D = AB + BC + CD + DE + EF
By putting the values then we get,
D = 4.2 + 1.4 + 2.1 + 0.7 + 0.7
D = 9.1 km
We can calculate the magnitude of the displaced objects by using the Pythagorean formula which is given there:
AF2 = AH2 + HF2
According to the given figure,
AH = 2.8
HF = 2.1
Now by using the formula,
AF2 = AH2 + HF2
Putting the values then we get
AF2 = ( 0.7 4)2 + ( 0.7 3)2
AF2 = (2.8) 2 + ( 2.1)2
AF2 = 7.84 + 4.41
AF2 = 12.25
Now taking square root on both sides
AF2 = 12.25
Then,
AF = 3.5 km
Example no 4:
The student goes to the school from house to school from point A to B then the initial point value A is 0 m and the final point value B is 9.5 m. then calculate the total displacement that can be displaced by the moving student.
To find:
Displacement = d = ?
Given:
A = initial point = xi = 0
B = final point = xf = 9.5
Formula:
Δx = xf - xi
Solution:
By putting the values in the formula
Δx = xf - xi
Δx = 9.5 - 0
Δx = + 9.5
Now we can also explore the nature of displacement its significance, graphical representation, their application in real life.
All vector quantities provide both magnitude and direction. So the displacement which also has a vector nature provides both magnitude and direction. In displacement the magnitude is equal to the straight path or line that is present between the two points starting or ending and the whole path is covered by the moving object. the direction of the displaced object is the line or path that starts from the initial or ends to the final point. By using the algebraic method or formulas for adding, subtracting, or multiplying rules we can also perform these operations on displacement. Simply we can add or subtract the displacement according to the vector's algebraic rules or methods. Vector adding is complex but we can do many other mathematical operations by using different methods or by following the vectors rules efficiently.
The displacement of the displaced objects can be calculated but it can depend upon their rectangular coordinate system or on the dimensions in which it can lie. Because to can use the simple displacement formula when we can calculate the displacement of displaced objects that can be lied on one dimension. But when we want to calculate the displacement of those displaced objects which can be lied on two or three dimensions we can use the vector rule or method for their subtraction. the formula which can be used to calculate the displacement are given there.
Displacement in one dimension:
The object can be moved in one direction or one dimension, Displacement is the distance that can be covered from the initial and the final point and it can easily calculated by subtracting the final value point from the initial value point. Their formula or equation is represented as:
d = xf – xi
Displacement in two dimensions:
The object can be displaced in two dimensions, Displacement is the distance that can be covered from the initial ( x1, y1) and the final point (x2, y2) along the axis or dimension of x and y. Their formula or equation is represented as:
d = ( x2 - x1) i + ( y2- y1) j
there,
i represented the unit vector on the x-dimension.
And,
j represented the unit vector on the y-dimension.
Displacement in three dimensions:
The object can be displaced in two dimensions, Displacement is the distance that can be covered from the initial ( x1, y1) and the final point (x2, y2) or the ( z1, z2) along the axis or dimension of x, y, and z. Their formula or equation is represented as:
d = ( x2 - x1) i + ( y2- y1) j + ( z2 - z1) k
there,
i represented the unit vector on the x-dimension.
j represented the unit vector on the y-dimension.
and,
k represented the unit vector on the z dimension.
Now the detailed discussion about the displacement in one dimension or two or more dimensions is given there:
When the object can be displaced in one dimension then there displacement is in a straight line so that's why the displacement value may be positive or negative because its direction can slo indicating the straight line motion in one dimension with a single axis x.
Let us consider whether the object can be displaced from the starting point or the initial point value is 4 m and the final or ending point value is 6 m then we can calculate the displacement by using the formula and they are given there:
d = xf – xi
By putting values
d = 6 m - 4 m
d = + 2 m
But if the object can move backward then the displacement value is negative then their initial value is 6 and the final value is 4 then it can be written as;
d = 4 m - 6 m
d = – 2 m
The displacement value is negative.
When the displacement lies in two or three dimensions then it can be represented in the form of a vector because in higher dimensions it is complex to represent the displacement in simple form so that's why it can be represented in multiple axes with vectors. These methods are mostly used to deal with the complex paths in which the objects are displaced.
Example of displaced objects in two dimensions:
The objects that are moving in the path have the starting point A, their initial point values are ( 2, 4) and the ending point values B are ( 3, 8) now find their displacement, magnitude, and direction also.
To find:
Displacement = d = ?
magnitude = d = ?
Direction = θ =?
Given:
x1, y1 = 2, 4
x2, y2 = 3, 8
Formula:
For displacement:
d = ( x2 - x1) i + ( y2- y1) j
For magnitude:
d = A2+ B2
For direction:
θ = tan-1(BA)
Solution:
Firstly we find the displacement by using the formula of displacement.
d = ( x2 - x1) i + ( y2- y1) j
By putting the values
d = ( 3 - 2 ) i + (8 - 4) j
d = 1 i + 4 j
Now use the formula of magnitude:
d = A2+ B2
By putting the values
d = 12 + 42
d = 1 + 16
d = 4.1
Now use the formula of direction:
θ = tan-1(BA)
By putting values
θ = tan-1(41)
Example of displaced objects in three dimensions:
The objects that are moving in the path have the starting point A, their initial point values are ( 2, 4, 2) and the ending point values B are ( 3, 8, 7) now find their displacement and magnitude.
To find:
Displacement = d = ?
magnitude = d = ?
Given:
x1, y1 = 2, 4
x2, y2 = 3, 8
z1 , z2 = 2, 7
Formula:
For displacement:
d = ( x2 - x1) i + ( y2- y1) j
For magnitude:
d = A2+ B2
Solution:
Firstly we find the displacement by using the formula of displacement.
d = ( x2 - x1) i + ( y2- y1) j + ( z2 - z1) k
By putting the values
d = ( 3 - 2 ) i + (8 - 4) j + (7 - 2 ) k
d = 1 i + 4 j + 5 k
Now use the formula of magnitude:
d = A2+ B2+ C2
By putting the values
d = 12 + 42 + 52
d = 1 + 16 + 25
d = 42
d = 6.5
For understanding the fundamental concept of relative motion, displacement concepts are basic. because in relative motion the displacement of the one-displaced object is assumed to the other relative motion. When the two objects are displaced or in continuous motion, then the displacement of one displaced object is relative to the other displaced object and they have the vector difference according to their displacement values.
For example:
Let us consider the two objects A and B are displaced at the path, their starting moving point values or the ending point values of object A are ( 2, 4) to ( 6, 8) and the starting and the ending point values for the object B are ( 0, 0 ) to (1, 4). They find the displacement of the object A relative to the object B.
To find:
drel = ?
d A = ?
d B = ?
Given:
A = (2, 4) (6, 8)
B = (0, 0 ) (1,4)
formula :
d A = xf – xi
d B = xf – xi
drel = d A – d B
Solution:
Firstly find the d A by using the formula
d A = xf – xi
By putting values
d A = (6-2) - (8-4)
d A= 4i + 8j
Then find the d B by using the formula
d B = xf – xi
d B = ( 1 -0 ) + ( 4 -0 )
d B = 1i + 4j
Then find the relative displacement by using the formula
drel = d A – d B
Now by putting the values
drel = (4i + 8j) - ( 1i + 4j)
drel = 3 i + 4j
Displacement can slo be represented in the form of a graph because they are vectors so that's why the vectors are complex and efficiently represented on the graphs.
In the graphical representation, the displacement is represented by putting the arrow on their starting or ending points. In the graphics; the representation of the displacement the slope represents the velocity of the displaced object.
In the given graphical representation of the displacement the straight line represents the positive velocity that is constant and does not change with time but in the next graphical representation, the curve represents that the velocity changes with time and they are not constant.
The graphical representation can be changed when the displaced object's starting or final values are different with different velocities this graph representation depends upon the velocity and the starting or ending point values of displaced values.
Concept of the displacement is used widely in many different fields because it is the fundamental concept that is used in the motion of objects so that's why they are widely used mostly in the different fields of physics. some important applications of displacement in the field of physics are given there:
Sports
Engineering
Medicine
Navigation
In the field of sports, displacement helps to analyze the performances and athlete's strategies because displacements help to increase the strength and ability of the athletes to cross the hurdles efficiently.
In the field of engineering, displacement concepts are the fundamental concept because when engineers design new vehicles and machine structures then they use displacement to analyze them. Displacement helps to analyze the function of the machines with specific parameters because when machines or vehicle start their motion they produce displacement so that's why firstly displacement can be analyzed.
In the field of medical sciences doctors and medical professionals study the motion of the different body parts of humans and animals through displacement. In biomechanics or the field of physical therapy, it is important to know about the motion of all body parts so displacement helps them to analyze or understand all motions of body parts efficiently.
In the field of navigation, the pilots locate their locations and sed the location to other because they analyze their location through their displacement. The sailors also locate their location with the help of the displacement vectors so that's why displacement vectors are essential and play a fundamental role in the field of navigation.
The complex vectors lie in different coordinate systems so the displacements can be analyzed in a cartesian system by following the vectors rule. Mostly the displacement can be analyzed in the special coordinate system, cartesian coordinate system, and mostly in the polar coordinate system. Some unique and specific types of problems according to vectors can be resolved through them. The details are given:
Spherical coordinate system
Cartesian coordinate system
Polar coordinate system
In the field of physics, displacement is the fundamental concept to understand efficiently the motion of the object and their description of the change in position of the moving object. The theoretical problems or different problems that can occur in the motion of the object can be solved efficiently after understanding the displacement crucially. The concept of displacement is the key or the basic concept to understanding displaced object movements in one dimension, navigation, medicine, engineering, or in numerous fields. After reading the whole article the reader can understand or use the displacement concept efficiently in numerous activities to understand the motion of the object.
Hi readers, I hope you are all well. In this post, we discuss the main topic, equilibrium. Equilibrium can play a fundamental role in the modern and different fields of science. In physics, engineering, or also in chemistry the concept of equilibrium describes and provides information about forces that can be applied to the system. In physics, the equilibrium describes the balanced forces and the torque that can be acted on the system or the body. if the body is at rest or in a motion state, equilibrium explains and provides information about the forces that can act in both types of situations persons.
Equilibrium plays a vital role in understanding the forces and the torque because equilibrium provides stability to the system, also it stabilizes the person who is in the state of rest or motion because in both conditions forces always act on it, so for their stabilization equilibrium plays an essential role.
In this post, we can discuss the equilibrium definitions, their types, first equilibrium condition, second equilibrium condition, mathematical expressions, their applications, examples, and related phenomena.
Equilibrium can be defined as:
"The state in which the body is in the state of balance, or the body is in the state of motion or rest with uniform velocity and no net change occurs on it."
Simply equilibrium is a state in which the system or the body is at the condition of balance under the action of forces but there is no net change occurring. Every system achieves equilibrium at some conditions because, without an equilibrium state, the system or the object can't do its work properly.
There are two main types of equilibrium which are common and they are given there:
Mechanical equilibrium
Thermal equilibrium
Their description is given there:
Mechanical equilibrium is the main type of equilibrium and it can be defined as:
In the system or an object mechanical equilibrium occurs when there is no force, torque net force, or acceleration acting on the object or the system.
Mechanical equilibrium occurs during the state of motion mostly. Mechanical equilibrium can be divided into two types which are also the main types of equilibrium and they are given below:
Static equilibrium
Dynamic equilibrium
These are the further divisions of the mechanical equilibrium Details are given there:
Static equilibrium can be defined as:
"static equilibrium can achieved by the body when the body is at rest and all forces which can act on the body including torque and acceleration sum is equal to zero."
Dynamic equilibrium can be defined as:
"Dynamic equilibrium can be achieved by the body when the body moves with a constant velocity and the all forces and torque which can be acted on the body their sum is equal to zero.”
Thermal equilibrium can be defined as:
"The two objects or the two or more systems can achieve the state of thermal dynamics when no exothermic or endothermic heat exchange occurs and with some condition or time both systems can be reached to the same temperature."
These are the major types of equilibrium but the types of equilibrium in physics are given there:
The types of equilibrium in physics with their uses and examples are given there:
Dynamic equilibrium
Radiative equilibrium
Thermal equilibrium
Static equilibrium
Chemical equilibrium
Their detailed definitions, mathematical expressions, formulas, and examples are given there:
Dynamic equilibrium can be achieved by the body when the body moves with a constant velocity and all forces and torque which can be acted on the body their sum are equal to zero. In dynamic equilibrium all forces which can be acted on the object are balanced. Mostly dynamic equilibrium can be used to understand or determine the objects that can be moved without acceleration.
As we know in dynamic equilibrium the all forces sum is equal to 0 then it can be written as:
F = 0
Or the sum of torque is also equal to zero, hence it can be written as:
𝛕 = 0
The body or object can be rotated around its axis with uniform angular velocity and no acceleration can be calculated or measured and all sum of forces that can be acted are 0
The car that can be moved with uniform linear velocity on a straight road, and not change their speed then the force which can be acted to maintain the road friction or air friction can be produced through the driving force which can be produced through the engines.
The paratrooper can also fly in the sky due to the dynamic equilibrium which stabilizes them to fly in the sky efficiently.
The airplane can also fly with a state of dynamic equilibrium because it can fly with constant speed and the weight and thrust force can balance the drag force by balancing their forces it can fly efficiently.
Radiative equilibrium is the state that can be achieved by the system or the object by absorbing the radiation or emitting the radiation at the same time and rate through equal emitting and absorbing radiation of the system and the object can achieved the dynamic equilibrium state efficently.
Rate of absorption of radiation = rate of emission of radiation
Stars can also maintain their lifecycle phase by emitting or absorbing radiation and achieving radioactive equilibrium. the stars produce energy through the nuclear fusion reaction and it can also radiate the absorbing energy into space and maintain gain radiative equilibrium.
The temperature or radiation can come from the sun and is maintained in the earth by radiative equilibrium because the earth absorbs all solar radiation and emits infrared radiation on the surface of the earth.
The two objects or two or more systems can achieve the state of thermal dynamics when no exothermic or endothermic heat exchange occurs and with some condition or time, both systems can reach the same temperature. thermal equilibrium concepts are essential in thermodynamics and also in their laws.
The temperature of the object and the system that can achieve the thermal equilibrium can represented through the symbol T.
No heat exchange occurs in the system when they achieve the thermal equilibrium then it can be written as:
T1 = T2 = T3=........ = Tn
For the two metal spoons, one is cold or the other is hot but if we check after some time then it can be observed that both metal spoons have the same temperature because both of them achieve thermal equilibrium.
If we leave the hot cup of tea or coffee in the room or open environment, then after some time we can observe that the hot cup of tea or coffee temperature becomes equal to the room or environment temperature,
Static equilibrium can achieved by the body when the body is at rest and all forces that can act on the body including torque and acceleration sum are equal to zero. In the static equilibrium state, the object always remains at rest so that's why the static equilibrium can be used to determine or understand those objects that can't move and always remain at rest. Simply static means rest so static equilibrium can only achieved by those objects or systems that can't be moved.
As we know in static equilibrium the all forces sum is equal to 0 then it can be written as:
F = 0
F represents the sum of all forces that can be acted on the body or object.
Or the sum of torque is also equal to zero, hence it can be written as:
𝛕 = 0
𝛕 represents the sum of all torques that can be acted on the body or object.
The book can be lying on the table or at rest, then the forces that can be acted on the table are maintained or normalized through the gravitational force that can be acted on the book which is lying on the table
The bridge that can be used for traffic is always in a static state, then it can maintain its static equilibrium by balancing the forces and the weight or load that act on the bridge.
In chemical reactions, chemical equilibrium can be achieved, when the forward and the reverse reaction rates are the same under the same conditions, and when the concentration of the products and the reactants can't be changed during the reaction then this state can be achieved. Mostly in chemistry, the chemical equilibrium can be used to determine or understand the concentration of reaction but in physics sometimes it can be used.
The rate of forward reaction = the rate of reverse reaction
The solution of the salt becomes in the equilibrium or saturation state when the rate of salt dissolution is equal to the rate of precipitation. But if we can provide the temperature to the solution of salt then we can change their equilibrium state also.
When we close the container in which hydrogen iodide solution is present, then this solution can achieve the equilibrium state easily because, in the closed container, we can't change the conditions and can't change the temperature or concentration of the reactant and product amount.
Two main conditions are essential for achieving the equilibrium state. If the object and the system can't follow these two main conditions then it can't achieve the equilibrium state. The two conditions for the equilibrium state are given there:
The first condition of equilibrium ( equilibrium of forces)
A second condition of equilibrium ( equilibrium of torque
in the previous post, we can discuss the first condition of equilibrium and now we can discuss the detail of the second condition of equilibrium.
The second condition of equilibrium can also referred to as the equilibrium of torque. According to the second condition, the all torque that can be acted on the body, their sum is always equal to zero. If the object or system follows this condition then it means that the body can't rotate around its axis and can't do the rotational motion.
The second condition of equilibrium is defined as:
The sum of all vector torque that can be acted on the object or the system is always equal to zero. Because this condition describes that the object or the system can't do the rotational motion around their axis.
𝛕 = 0
𝛕 represented the sum of all torque that can be acted on the body.
As we know the torque is equal to the position vector or the distance from the axis of rotation and the vector product of force F and the sin θ are the angle between the r and f. The it can be mathematically represented as:
𝛕 = r F sin θ
Then if the sum of all torque is equal to zero it proves that the body which can be moved with rotational motion is at the equilibrium state and all forces have become zero and it can be written as:
𝛕 = 0
When all forces that can be acted on the object in one plane or are coplanar then we can apply the condition of equilibrium we stabilized or maintained them.
All forces can be passed through one point which is the line of action and the body moves around into its axis within the line of action.
By choosing the axis we can calculate the torque efficiently and the position of the object and the position of the axis is arbitrary.
The second condition of equilibrium or the equilibrium of forces is essentially used to determine or understand those systems or objects that can do the rotational motion.
This condition is the base in the field of dynamics because in this field we can deal with different types of motion. In mechanics, it can help to analyze the structures and the components that can be used in the designing of the system which can do the rotational motion, and also analyze how they achieve the equilibrium state by balancing the forces and the load which can be acted on them.
The seesaw can pivot in the center. Two children with different weights sit on both sides but they can show the equilibrium state when the torque that can be acted on is equal to zero by balancing the load or forces that can be acted on the swing.
The ladder can stand with the support of the wall. The ladder may fall but if it becomes at the equilibrium state by balancing the forces then it can't fall.
The second condition of equilibrium can follow many different fields of science and it can be used in many different applications that can be used in daily life some explanations are given there:
Every day situations
Structural engineering
Mechanical system
In our daily lives, the second condition of equilibrium can be used to balance or stabilize many different things. All systems can be managed or stabilized due to equilibrium. For instance, the picture can be hung with the hook and the weight of the picture or sign can be balanced through the hook. Torque can also be produced by the picture and it can be balanced by following the second condition of equilibrium.
In the field of engineering, engineers can design or choose the components that can manage the rotational motion with equilibrium and manage all forces or torque that can be acted on it. Engineers always prefer to choose those components that can efficiently work and remain in the equilibrium state. For instance, the cantilever beam can be designed by the engineers, they calculate the all torque that can be acted on it and then also analyze that they can able to bear the load or ensure that the beam is at an equilibrium state or not move around their axis to produced torque.
The second equilibrium can be used in the field of mechanics in which the components are designed to work properly without error. The second condition of equilibrium can also used for checking the proper functioning of the machines and also for their safety and for increasing their efficacy to do work properly. For instance, the gear systems that are used in the vehicles are designed by the engineers, they can be designed by ensuring the components can balance all forces and the torque must be equal to 0.
With time and with the development of modern science and technology equilibrium can be used in many different new topics with new concepts and ideas that can be presented through modern research. Some modern concepts and ideas about the second condition of the equilibrium are given there:
Equilibrium in the quantum system
Metastable equilibrium
Equilibrium in the dynamic systems
The second condition of the equilibrium can also now be used in the quantum system because in the quantum system the probabilities and managed or stabilized efficiently. The superpositions and the quantum tunneling can also be understood or determined through the second condition of the equilibrium. For instance, the electrons that can distributed in the conductors, the energy, and the distribution of the electrons can be managed or stabilized by using or following the conditions of the equilibrium.
The second condition of equilibrium can be used in the metastable, in this, the larger or smaller distribution can be managed or stabilized efficiently. For instance, the pencils that we can use can also be balanced on the tip, but if a small disturbance occurs the pencil can fall and distribute the equilibrium state easily.
In the dynamic system, the equilibrium occurs when it follows the second condition of the equilibrium. If we understand the equilibrium of torque then we can analyze or stabilize all control systems or dynamic systems. For instance, the satellites can be moved around their axis, and in the orbit, their stabilization can be managed by following the second condition. Because the second condition of the equilibrium maintained to move in orbit or doesn't allow them to move irregularly in the other orbits.
The advanced topics in which the second condition of equilibrium is used are given there:
Equilibrium in elastic system
Equilibrium in the three-dimension
Multiple forces equilibrium in the system
Some practical examples in which the second condition of equilibrium is used are given there:
Aerospace engineering
Architectures
Building designs
Robotics
Automotive engineering
The equilibrium of torque, which is also referred to as the second condition of the equilibrium is the essential or fundamental concept in the dynamics or mechanics in which the system and the object can do the rotational motion. If we can apply the second condition of equilibrium we can stabilize the different applications in daily life or mechanics. In the era of the modern sciences, equilibrium is essential in every system for working properly and for better output efficiency. by understanding this article or post or understanding the second condition of equilibrium it is easy to balance the objects in the physical world and also in the major fields of science.
Hi readers, I hope you are all well. In this post, we discuss the main topic, equilibrium. Equilibrium can play a fundamental role in the modern and different fields of science. In physics, engineering, or also in chemistry the concept of equilibrium describes and provides information about forces that can be applied to the system. In physics, the equilibrium describes the balanced forces and the torque that can be acted on the system or the body. if the body is at rest or in a motion state, equilibrium explains and provides information about the forces that can act in both types of situations persons.
Equilibrium plays a vital role in understanding the forces and the torque because equilibrium provides stability to the system, also it stabilizes the person who is in the state of rest or motion because in both conditions forces always act on it, so for their stabilization equilibrium plays an essential role.
In this post, we can discuss the equilibrium definitions, their types, first equilibrium condition, second equilibrium condition, mathematical expressions, their applications, examples, and related phenomena.
Equilibrium can be defined as:
"The state in which the body is in the state of balance, or the body is in the state of motion or rest with uniform velocity and no net change occurs on it."
Simply equilibrium is a state in which the system or the body is at the condition of balance under the action of forces but there is no net change occurring. Every system achieves equilibrium at some conditions because, without an equilibrium state, the system or the object can't do its work properly.
There are two main types of equilibrium which are common and they are given there:
Mechanical equilibrium
Thermal equilibrium
Their description is given there:
Mechanical equilibrium is the main type of equilibrium and it can be defined as:
In the system or an object mechanical equilibrium occurs when there is no force, torque net force, or acceleration acting on the object or the system.
Mechanical equilibrium occurs during the state of motion mostly. Mechanical equilibrium can be divided into two types which are also the main types of equilibrium and they are given below:
Static equilibrium
Dynamic equilibrium
These are the further divisions of the mechanical equilibrium Details are given there:
Static equilibrium can be defined as:
"static equilibrium can achieved by the body when the body is at rest and all forces which can act on the body including torque and acceleration sum is equal to zero."
Dynamic equilibrium can be defined as:
"Dynamic equilibrium can be achieved by the body when the body moves with a constant velocity and the all forces and torque which can be acted on the body their sum is equal to zero."
Thermal equilibrium can be defined as:
"The two objects or the two or more systems can achieve the state of thermal dynamics when no exothermic or endothermic heat exchange occurs and with some condition or time both systems can be reached to the same temperature.”
These are the major types of equilibrium but the types of equilibrium in physics are given there:
The types of equilibrium in physics with their uses and examples are given there:
Dynamic equilibrium
Radiative equilibrium
Thermal equilibrium
Static equilibrium
Chemical equilibrium
Their detailed definitions, mathematical expressions, formulas, and examples are given there:
Dynamic equilibrium can be achieved by the body when the body moves with a constant velocity and all forces and torque which can be acted on the body their sum are equal to zero. In dynamic equilibrium all forces which can be acted on the object are balanced. Mostly dynamic equilibrium can be used to understand or determine the objects that can be moved without acceleration.
As we know in dynamic equilibrium the all forces sum is equal to 0 then it can be written as:
F = 0
Or the sum of torque is also equal to zero, hence it can be written as:
𝛕 = 0
The body or object can be rotated around its axis with uniform angular velocity and no acceleration can be calculated or measured and all sum of forces that can be acted are 0
The car that can be moved with uniform linear velocity on a straight road, and not change their speed then the force which can be acted to maintain the road friction or air friction can be produced through the driving force which can be produced through the engines.
The paratrooper can also fly in the sky due to the dynamic equilibrium which stabilizes them to fly in the sky efficiently.
The airplane can also fly with a state of dynamic equilibrium because it can fly with constant speed and the weight and thrust force can balance the drag force by balancing their forces it can fly efficiently.
Radiative equilibrium is the state that can be achieved by the system or the object by absorbing the radiation or emitting the radiation at the same time and rate through equal emitting and absorbing radiation of the system and the object can achieved the dynamic equilibrium state efficently.
Rate of absorption of radiation = rate of emission of radiation
Stars can also maintain their lifecycle phase by emitting or absorbing radiation and achieving radioactive equilibrium. the stars produce energy through the nuclear fusion reaction and it can also radiate the absorbing energy into space and maintain gain radiative equilibrium.
The temperature or radiation can come from the sun and is maintained in the earth by radiative equilibrium because the earth absorbs all solar radiation and emits infrared radiation on the surface of the earth.
The two objects or two or more systems can achieve the state of thermal dynamics when no exothermic or endothermic heat exchange occurs and with some condition or time, both systems can reach the same temperature. thermal equilibrium concepts are essential in thermodynamics and also in their laws.
The temperature of the object and the system that can achieve the thermal equilibrium can represented through the symbol T.
No heat exchange occurs in the system when they achieve the thermal equilibrium then it can be written as:
T1 = T2 = T3=........ = Tn
For the two metal spoons, one is cold or the other is hot but if we check after some time then it can be observed that both metal spoons have the same temperature because both of them achieve thermal equilibrium.
If we leave the hot cup of tea or coffee in the room or open environment, then after some time we can observe that the hot cup of tea or coffee temperature becomes equal to the room or environment temperature,
static equilibrium can achieved by the body when the body is at rest and all forces that can act on the body including torque and acceleration sum are equal to zero. In the static equilibrium state, the object always remains at rest so that's why the static equilibrium can be used to determine or understand those objects that can't move and always remain at rest. Simply static means rest so static equilibrium can only achieved by those objects or systems that can't be moved.
As we know in static equilibrium the all forces sum is equal to 0 then it can be written as:
F = 0
F represents the sum of all forces that can be acted on the body or object.
Or the sum of torque is also equal to zero, hence it can be written as:
𝛕 = 0
𝛕 represents the sum of all torques that can be acted on the body or object.
The book can be lying on the table or at rest, then the forces that can be acted on the table are maintained or normalized through the gravitational force that can be acted on the book which is lying on the table
The bridge that can be used for traffic is always in a static state, then it can maintain its static equilibrium by balancing the forces and the weight or load that act on the bridge.
In chemical reactions, chemical equilibrium can be achieved, when the forward and the reverse reaction rates are the same under the same conditions, and when the concentration of the products and the reactants can't be changed during the reaction then this state can be achieved. Mostly in chemistry, the chemical equilibrium can be used to determine or understand the concentration of reaction but in physics sometimes it can be used.
The rate of forward reaction = the rate of reverse reaction
The solution of the salt becomes in the equilibrium or saturation state when the rate of salt dissolution is equal to the rate of precipitation. But if we can provide the temperature to the solution of salt then we can change their equilibrium state also.
When we close the container in which hydrogen iodide solution is present, then this solution can achieve the equilibrium state easily because, in the closed container, we can't change the conditions and can't change the temperature or concentration of the reactant and product amount.
Two main conditions are essential for achieving the equilibrium state. If the object and the system can't follow these two main conditions then it can't achieve the equilibrium state. The two conditions for the equilibrium state are given there:
The first condition of equilibrium ( equilibrium of forces)
A second condition of equilibrium ( equilibrium of torque
Details of the first condition of equilibrium are given there:
The first condition of equilibrium is also known or referred to as the equilibrium of forces because in this condition all forces that act on the body or the object must equal to zero. If the system or the object can't follow this condition then it never can achieve the equilibrium state.
The first condition of equilibrium or the equilibrium of forces can be defined as:
The sum of all vector forces that can act on the object or any system externally is always equal or must equal to zero. Mathematically it can be written as:
F = 0
This equation proves or shows that the sum of all forces is equal to zero so that is why the object that is at rest or motion has not been accelerated because the object is in a uniform state of motion or the rest.
If the system which can follow the first condition of equilibrium to achieve the state of equilibrium can lie in the two dimensions then it can be written as:
Fx = 0
This equation or formula represented the sum of all forces in the x direction.
Fy = 0
This equation or formula represented the sum of all forces in the y direction.
Both equations can be used in the two dimension system. But if we balanced the three-dimensional system then the forces are directed on three axes x, y, and z and it can be written as:
Fx = 0
This equation or formula represented the sum of all forces in the x direction.
Fy = 0
This equation or formula represented the sum of all forces in the y direction.
Fz = 0
This equation or formula represented the sum of all forces in the z-direction.
If the forces that can be acted on the object or system are taken in the right direction then these forces are positive.
If the forces that can be acted on the object or system are taken in the left direction then these forces are positive.
If the forces that can be acted on the object or system are taken in the upward direction then these forces are positive.
If the forces that can be acted on the object or system are taken in the downward direction then these forces are positive.
If the forces that can be acted on the object or system are common in plane then these forces are termed as the coplanar.
All stationary and static systems can be analyzed or understood through the first condition of equilibrium. The branch of mechanics in physics, the concept of equilibrium can be discussed in statics principle in which we can deal with or study the forces that can act on a stationary system or object. The equilibrium of forces is fundamentally used for understanding or analyzing the
Structural analysis
Mechanical analysis or designs
Everyday applications
Every day we can use many different objects in which the equilibrium can be seen. In common conditions like hanging the objects and placing different things one by one or balancing them. In hanging or balancing things can be done by the equilibrium so we can also understand the equilibrium interactions in our daily life.
In engineering or physics, we can understand the structures of the system through which the balance can be maintained like the bridges, roads, and buildings that can carry the load stabilized due to the equilibrium condition by balancing the all forces which can act upon it.
Ensuring the stability of buildings, bridges, and other structures under various loads.
In the field of mechanics, equilibrium conditions can be followed to design the efficient machine or their designs. In designing the machines the components are chosen that can stand with the forces that can act upon it and maintain them efficiently at the rest or in motion also.
The equilibrium in the static sign can be hung on the wall.
Let us consider the sign that we can hang on the wall with two ropes so they have different angles. The angle that can be made by the ropes during hanging is approximately 30° or 45°. The weight of the sign that can be hanged is 100N. Now we can find or determine the tension that can be produced in ropes for balancing the sign on the wall.
Now let the T1 tension for the first rope which can make an angle of 45° and T2 the tension for the second rope which can make an angle of 55°. For achieving the equilibrium state all forces that can act on the sign in the x and y direction, their sum is must be equal to 0.
The forces that can act in the x direction:
T1 cos (30) = T2 cos (45)
The forces which can act on the y direction:
T1 sin ( 30) + T2 sin ( 45) = 100
Now we can solve both of these equations and write as:
T1 cos (30) = T2 cos (45)
T1 32 = T2 12
And then it can be written as:
T1 = T2 23 12
Then,
T1 = T2 26
T1 = T2 23 ……. (i) equation
Now we can solve the equation in which force can be acted in the direction of y and it can be written as;
T1 sin ( 30) + T2 sin ( 45) = 100
T1 12 + T2 22 = 100
Now we can put the value of T1 in the given equation and write it as:
(T2 23 ) 12 + T2 22 = 100
T2 (23 . 12 + 22 ) =100
Then,
T2 ( 22 3 + 22) =100
T2 ( 2 (1 + 3)23) = 100
T2 = 200 32 (1 + 3)
Then,
T2 = 73. 2 N
Equilibrium of forces or the first condition of equilibrium can be used in many different fields because it can help to describe the forces or to maintain the system some applications of equilibrium of forces are given there:
Structural engineering
Everyday situations
Mechanical systems
In the field of engineering, where bridges buildings, and many other machines can be designed their equilibrium of forces is essentially used because it can ensure that the design or the components we can use have the ability to carry the load to maintain their balance. And can't collapse or be destroyed due to the imbalance of force or weight. Engineers study or examine whether the components that they can use are efficient or attain equilibrium efficiently or not.
Example:
The engineer can design the bridge through which the traffic can be passed, they use the best components and materials that can manage the forces carry the load efficiently, and attain the equilibrium state without deforming or collapsing.
Every day we can use many different objects in which the equilibrium can be seen. In common conditions like hanging the objects and placing different things one by one or balancing them. In hanging or balancing things can be done by the equilibrium so we can also understand the equilibrium interactions in our daily life.
Example:
When we hang the pictures or any sign on the wall with ropes then the ropes can manage the tension and all forces that can be acted on them and stabilize them to hang the wall without falling. The all forces which can be acted on the ropes and the sign or picture their sum are always equal to zero. And after neutralizing the forces they can achieve stabilization or equilibrium.
In the field of mechanics, equilibrium conditions can be followed to design the efficient machine or their designs. In designing the machines the components are chosen that can stand with the forces that can act upon it and maintain them efficiently at the rest or in motion also. In the mechanical field, engineers choose the components very precisely to maintain the equilibrium state.
Examples:
When engineers make heavy machines or vehicles like cranes then they check or ensure that the forces that can be acted on the crane arm or the load that can be placed on it are capable of bearing it or not. After ensuring these components' ability they can be allowed to use it or made efficient and heavy machinery.
In the modern era of science and technology, the equilibrium of forces can be used in many new fields according to their need. Some new topics in which the equilibrium of forces can be used are given there:
Equilibrium in elastic system
Equilibrium in three-dimension
Equilibrium in the system with friction
In the elastic system, the equilibrium can be used because when we stretch the spring it can be restored and maintained by balancing the forces that can be acted on it. Equilibrium is essential in all systems and objects because without equilibrium the system can't work properly.
In two dimensions equilibrium can easily attained but now in three dimensions equilibrium must attained by balancing the forces and using the best components in it. For instance, cranes and the tripod or the tables can stand in equilibrium and manage the forces acting upon it in the three dimensions x, y, and z efficiently.
In equilibrium problems, friction can play a vital role. Because when the system interacts with the surface it is obvious that friction can be produced so when we calculate the system's net force for their equilibrium the force of friction can also be calculated because sometimes, friction is the restriction in attaining the equilibrium set. For instance, if we place a block on the surface or the plane then if we want to stabilize the block at the right position it is a must to calculate the all forces that can act on the block along with the force of friction.
Some practical examples in which the equilibrium of forces are commonly used to maintain their work or maintain their output efficiency or control on them are given there:
Robotics
Automotive engineering
Architecture
Building designing
Aerospace engineering
In the era of modern science and technology, the first condition of equilibrium can play a fundamental role in different fields like physics, mechanics, chemicals, or engineering because it can provide basic information and help to understand all static systems and objects efficiently. By understanding this condition of equilibrium we can achieve the system balanced without producing the error or acceleration. Through their application in many new fields, we can easily understand them because this condition of equilibrium can be used in our daily life situations. The equilibrium of forces is easy to understand because in this the sum of forces is always equal to zero or the forces include the normal force, gravitational force, or maybe the frictional force. But with time it can be more commonly used in the field of science. Equilibrium of forces is the fundamental concept in the field of static or dynamics
Hi readers, I hope you are all well. In the previous articles, we discussed different basic physics topics like vectors and others and now we can discuss the major topic torque in this article. Torque can also be called the rotational force or moment of the force. In physics, mechanics, and engineering torque plays a vital role and has a fundamental concept in these fields. Torque is essential to understand or describe the ability of the force of the object to rotate around its axis. The object moves with the linear force but when the object moves around its axis, some pivot point, or the fulcrum with some force then this rotational force of the object is termed as the torque.
The word torque is derived from the Latin word "torquere" which means turning or twisting. Torque describes both magnitude and direction so that is why it is a vector quantity because only vector quantities describe both magnitude and direction. Rotational dynamics The branch of dynamics can be understood through torque because the rotational force of the object is only described through torque. Torque can be represented through the symbol 𝛕. Torque can also be represented through the M. Now we can start our detailed discussion about torque, its definition, relation with other quantities, their mathematical expression, significance, applications, examples, and many other phenomena.
The idea and concept of the torque was presented by the Archimedes. The use of the lever instrument was studied by the Archmedies and when they studied them then the idea of torques was first described by him. But the term or word torque was first advised by the great scientist James Thomson. Then the experiment was done by scientist P. Thompson in the same year when the idea was described and the experiment was written in the Dynomo electric machinery book in their first edition in the 18th century, 1884.
After this, the great and famous scientist Newton can present or describe the force through which the body moves. So according to their definition when the linear force acts on the moving object and any twist occurs around their axis and changes the force of the body from linear to rotation then it is termed as the torque. For instance, the screwdriver always rotates around its axis As another example of torque the seesaw swing which can be off and on on the groping due to the imbalance of the torque.
Now torque can be used or referred to in many different fields of science to understand the rotation force of a body at the given pivot point or around its axis. Mainly the concept of the moment of force can be described in the early 18th century in 1811, but it can print in the late 18th century and the then torque is also referred to as the moment of force, and the rotational force.
Torque can be defined as:
"the body which can move around their axis with the turning force which can be produced by the body is known as torque."
Torque can also defined as:
The magnitude of the perpendicular distance of the body from the axis of the rotation or the magnitude of the force, product of these both magnitudes are termed as the torque or moment of force:
The torque of the moment of force can also be defined as:
"the cross or the vector product of the radius or the position vector r and the vector force F".
Torque can be written in mathematical form as:
𝛕 = r F
There,
𝛕 represented the torque and the magnitude of the torque was represented through 𝛕.
r represented the position vector and the magnitude of the position vector is represented through the r. position vector is the distance which can be measured by the torque through the point where the force is applied to the axis of rotation.
F represents the force and the magnitude of the force is represented through the F.and the force is perpendicular to the position vector r and represented by the symbol ⊥ .
The magnitude of the torque can be written as:
𝛕 = r F sin
There,
𝛕 denotes the magnitude of the torque.
r denotes the magnitude of the position vector.
denotes the cross or the vector product between the two vectors
F denotes the magnitude of the force.
θ is the angle that is present between the position vector and the force, both of these are the vector quantities.
The direction of the torque can be represented or described through the right-hand rule. Because this rule can efficiently represent the direction of both of these vectors r and the F. According to the right-hand rule, the thumb can represent the torque 𝛕 which is the product of two different vector quantities, then the fingers of the right hand represent the direction of the position vector r and the curl fingers represent the direction of force F.
Torque can depend on the two major factors which are given there:
Moment arm
Magnitude of the force
Both of these factors are the major factors that can directly affect the torque. The moment arm is simply defined as the perpendicular distance of the body from the line of action to the axis of rotation. And these measurements of the moment arm with the force are simply termed the torque.
The torque can show the relationship with many other physical quantities like power, angular momentum, and energy. Mathematical derivation which can show the relationship of torque with other quantities in detail is given there:
The term torque can also be described or understood by using the law of conservation of energy. Because when the body moves with some force then it can cover some distance so they are the mechanical work that can be done by the body. In the angular displacement when the torque acts then the body is in the condition of doing work. The turning force acts on the body around its axis which is fixed with the center of mass so that's why mathematically work with torque can be expressed or written as:
W = θ1θ2 𝛕 d θ
There,
W represented the work that is done by torque.
𝛕 represents the torque through work that can be done
d is the angular displacement in which the torque or turning force can be acted
θ1 or they θ2 are the angle between the initial angular displacement point to the final angular displacement point.
Work energy principal:
The torque can show the relation with the energy or the work according to tyo the work-energy principle in which the work or energy can be changed into rotational kinetic energy on the body and it can be represented or written as:
Er = 12 I ω2
There,
Er represent the rotational kinetic energy
I represented the inertia of the body
ω2represented the angular speed of the body through which they can cover the angular momentum.
Power can also defined in the form of energy or work with unit joule. but the relation of the power with torque is the work that can be done in the unit of time and it can be written as:
P = 𝛕 ω
There,
P represents power.
𝛕 represents the torque.
ω represents the angular velocity through which the body does work.
represented the dot or the scalar product between the two vector quantities torque and angular velocity.
According to this mathematical expression or equation, it can be shown that the torque and the angular velocity scalar or dot product can give the output of the power. the torque in the relationship of the power was dependent upon the angular velocity or the speed. But the torque does not depend on the velocity decreased or maybe increased but it only depends upon the angular velocity. The force also depends upon the velocity of the object on the acceleration or the speed.
The work that can be done on the body when the random variable force acts on the body in the liner displacement or the force which can act on the body with the respect of elemental displacement then it can be written as:
W = s1s2 F . ds ……. (i) equation
There,
S1 and S2 are the initial and the final linear displacements that can be covered by the object during the work.
F represented the force
ds represented the elemental linear displacement.
So,
The elemental displacement ds are also equal to the cross or vector product of the radius and the angular displacement and written as:
ds = d θ r
There,
r represented the radius
d θ represented the angular momentum
Now put the value of the ds in the equation (i)
W = s1s2 F . d θ r
Now as shown in the equation the triple scalar product integers are shown and it can be also written as:
F . d θ r = F r . d θ
If we know that the radius or the angular momentum with force is equal to the torque then it can be written as:
W = s1s2 𝛕. d θ
But if both quantities, torque and angular momentum can lie in the same direction then the angle between them is cos, and their magnitude can be written as the:
= 𝛕. d θ
= r . d θ cos0
= 𝛕 d θ
Then it can be written as:
W = s1s2 𝛕 d θ
Angular momentum which can be acted on the body can be determined through the torque that can act on it and it can be written as:
𝛕 = dLdt
There,
𝛕 represented the torque
L represents the angular momentum
And the t represented the time with the displacement.
Or the angular momentum is also equal to the inertia of the moment and the angular speed and it can be written as:
L = I ω
I represented the moment of inertia and the w represented the angular speed.
And the moment of inertia I is also equal or written as:
I = m r2
Then the total net torque can be written as:
𝛕net = I1w1e1 + I2w2e2 + I3w3e3 + I1w1 de1dt + I2w2 de2dt + I3w3de3dt
𝛕net= Iw + w (Iw)
Then it can also be written as:
deidt = w ei
This equation can be used for newtons law but in some problems, there are only inertia and angular momentum then in a simple way, we can write them as the:
𝛕 = I a
There,
𝛕 represented the torque, I represented the inertia, and the a = w represented the angular velocity or the speed.
This equation can also be called the Newton's second law.
Simply, the angular momentum for a single particle can be defined or written as:
L = r p
There,
L represented the angular momentum of the single particle
r represented the position vector
p represented the linear momentum of the single particle.
But when we can write the angular momentum for time then mathematically it can be written as:
dLdt = dpdt r + drdt p
As shown in the given mathematical equation when we can split the equation into its components and then we can use the product rule of vector because the force is represented the rate of change in the momentumdpdt and the drdt change in the position of the quantity is represented through the velocity symbol v. then it can be written as:
dLdt= F r + v p
There,
V represented the velocity and the F represented the force. Now as shown in the given equation both vector quantities velocity v and the angular momentum p are parallel to each other so they are equal to zero 0 and it can be written as:
dLdt= F r
Now as shown in the given mathematical equation force and the position vector are equal to the torque and when we apply the Newton law then it can be written as:
𝛕net = dLdt= F r
Now through this equation, it can be proved that the torque has a significant relationship with the angular momentum of the single particle. This mathematical equation is the generalized proof of the torque and the angular momentum along the mass.
The units, symbols, and the dimensions of the torque are given there:
For the quantity torque, many units can be used but some major units that can used to express the torque are given there:
Nm ( newton meter)
Dyn . cm ( dyne - centimeter) This unit can be used in the CSG system to express the torque.
Pound foot represented by ( Ibf- ft)
Pound inch, this unit can be used to measure the small torque measurements and represented by ( Ibf- in).
Foot-pound can be represented through Ib- ft.
Like the foot-pound, the torque can also be represented through inch-pound and represented through the in-Ib.
The SI unit of the torque can be written as:
kg . m2 . s-2 ( kilogram meter square per second square)
Dimension of the unit torque is written as:
ML2T-2
Torque can be calculated and depends upon some major factors but some major formulas that are used to calculate the torque for the single force and for the multiple force are given there:
When a single force applies or acts on the body with some distance from their fixed axis of rotation, and the force that acts on the body is perpendicular to the position vector r, then the torque can be written as:
𝛕 = r F
But if the force is not perpendicular to the position vector r then it can be written as:
𝛕 = r F sin θ
there,
θ can represent the angle between the force and the position vector.
When the multiple forces act on the body or object then the torque can be calculated through their vector sum and written as:
𝛕net = 𝛕i
There,
𝛕net represented the total sum of the torque that can act on the body with multiple forces.
and,
𝛕i = ri Fi
Now we can calculate the torque that can act on the rigid body.
As shown in the figure, let us consider the rigid body. According to this figure, the force F acts on the object at the point p, r is the position vector according to the point p, and the angle between the force and the position vector r is represented through the Now we can calculate their torque by their resolution.
In the above figure, we can see the components of the vector according to the rectangular components and then it can be written as:
F cos θ = this is the component of the force in the rectangular component which can act in the direction of the position vector r.
F sin θ = this is the component of the force in the rectangular component which can be perpendicular to the position vector r.
As shown in the above equation, the F cos θ and their line of action can pass through the point O, and this rectangular component due to the line of action becomes zero 0. So that is why the force that can be acted on the body is equal to the F sin θ which can produce the torque and it can be written as:
𝛕 = r( F sin θ)
𝛕 = r F sin θ
Or when we can write with their magnitude or in vector form then it is as:
𝛕 = r F sin θ n
Also, it can be written as:
𝛕 = r F
After the component of force then we can write the rectangular component of the position vector r. It can be written there as:
r cos θ = the component of the position vector r along with the direction of the force.
r sin θ = the component of the position vector r perpendicular to the vector force F.
When the torque is produced due to the force, in this case, we can write the torque as:
𝛕 = l F
there,
L is equal to the moment arm and it can be written as:
l = moment arm = r sin θ
So it can be written as
𝛕 = ( r sin θ) F
𝛕 = r F sin θ
In the magnitude of the vector form it can be written as:
𝛕 = r F sin θ n
Also, it can be written as:
𝛕 = r F
A wrench ( spanner) is used to tighten the nut. The spanner moves around its axis of rotation with some force so it can produce the torque.
The swing seesaw can move up and down on the ground due to the imbalance of the torque because its center is fixed so it can move at the line of action and produce the imbalance torque.
For the rotational motion of the body, torque is the major or counterpart of the force that can act on the moving object.
The body can be moved through the linear motion and the angular motion in which force can be acted, the force is the same as the torque.
The linear acceleration can be described or determined through force and the angular acceleration can be determined or described through the torque.
The torque will be positive if the rotational motion of the object occurs in the anti-clockwise direction, and the torque will be negative if the rotational motion of the object occurs in the clockwise direction.
Torque can play a very fundamental role in many different fields some of their applications in different fields with detail are given there:
Robotics
Mechanical system
Sports and biomechanics
Structural engineering
Their detail is given there:
Torque plays a vital role in the robotic system. Because by the help or understanding of the torque the movement of the robots through their arms or joints is controlled efficiently. Torque helps to determine or describe the force that can act on the joints of the robot through which they can do rotational motion and help to control all movements of the robots precisely or efficiently.
Actuators: actuators are the systems in the robots that can convert the energy to the motion through which the robots can be moved. The functions of the robots are controlled efficiently through the torque. Torque also helps to choose the best and appropriate systems and actuators for making the robotic system accurate or precise.
Robotic arm: the joints are moved through the rotational force which is torque so that's why all joints or arms of the robot have some specific torque value through which they can move or complete their specific movement efficiently.
In the field of mechanics which is the branch of physics, torque are fundamental concept for designing engines or gears, and turbines or generators. Because in these given systems the engines can work with some force which is termed as torque and it can be moved or run through the rotational motion, the efficacy and the output of these systems can be measured also through torque.
Gear system: the efficient and precise gear system can be designed by determining the torque because the torque can be transmitted to the near components through which the system can run efficiently.
Engines: in the mechanical system the engine's efficiency and ability can be determined through the torque. Because the torque describes the ability and the capacity of the engine's acceleration and the capacity to carry the load. So to make efficient and design efficient engines torque can be used.
Torque can play an essential role or help to determine or analyze the movements of the human joints because the joints can do rotational motion. Through torque athletic performance can also be improved or managed. Torque also helps to understand which muscles can apply the force on the joint to do movements and through which muscle we can control the movement and prevent injury chances.
Injury prevention: by understanding the torque detail we can design the many equipment that can help to decrease the risk of injury and also prevent the human or athlete from injury. Torque also helps to understand which muscles can apply the force on the joint to do movements and through which muscle we can control the movement and prevent injury chances.
Athletic performance: the torque is produced by the muscle and these forces act on the joint through the movements that occur, by understanding the muscles that produce torque helps to improve performance, and with training the athlete can maintain or improve their skills efficiently.
The structure stability and the system ability can be determined through the torque. In structural engineering, bridges, buildings, and other designs are made after understanding the torque because it can help to understand the ability of the object to carry the load.
Bridges: when the engineers are designed to make the bridges they determine the torque because it can help them to find or observe whether the bridges can carry the load and are safe to use or not.
Torque can be used in many different advanced topics because the concept of torque can be presented in the 18th century and with time it can be used in many different fields some are given there:
Torque in electromagnetic systems:
Generators:
Electric motors:
Precession:
Gyroscopic effects:
Gyroscopes:
Measurements of torque are very essential so, to measure the torque efficiently and take precise and accurate measurements many different instruments and devices are used some of these are given there:
Dynamometers are the instruments that are used to measure the efficiency of the engines and the output of the engines with torque efficiently. Many types of dynamometers are used to measure these torque measurements some are given there:
Chassis dynamometer
Engine dynamometer
Used to measure the torque or also tighten the nuts through the wrenches by applying force on it.
Torque sensors which are also termed torque transducers, are instruments or devices that can be used to measure the torque of all rotating objects or systems. Torque sensors are mostly used in the automotive to control or monitor the torque. some types are given there:
Rotary torque sensor:
Strain gauge sensor:
Torque plays a vital role in different fields of science. Torque made the relationship and described the relationship between the linear motion and the rotational motion efficiently. Through understanding the torque we can also understand how the force acts on the body and how the object can be moved around its axis and do the rotational motion efficiently. Torque can be used in our daily life when we used the screw gauge, wrench, and others then they do the rotational motion and produced the torque efficiently. In the era of modern science, torque plays a fundamental role in simplifying complex problems efficiently. After understanding and reading their definition, and mathematical relationship with other physical quantities we can achieve great knowledge about this ubiquitous force that plays an essential role in the physical world.
Hi, friends I hope you are all well and doing the best in your fields. Today we will discuss the cross or the vector product. In the previous article, we discussed vector quantities, scalar quantities, and the scalar or dot product with their properties, and applications in different fields of science now we can talk about the cross or vector quantities in detail because vectors are used in mathematics, physics, engineering or many other fields. Algebraic operations can also be solved by using the vectors. Vectors are widely used because they can provide the magnitude and direction of a quantity.
The vector product is also known or termed as the cross product. Vector product or in the cross-product are binary vectors or these vectors are perpendicular to each other in the three-dimensional plane. Generally, the cross or the vector product can solve complex algebraic operations like torque, magnetic force, and angular momentum. The mathematics which is the field of science, the cross or the vector product can usually represent the product of the given area with the direction where the two vectors are placed in the Euclidean space or the three-dimensional Cartesian plane. The scalar or the dot product can be represented by the sign or symbol () but the cross or the vector product can be represented by the symbol which is termed a cross. The scalar or the dot product is different from the cross or the vector product because the scalar product can be also termed or used for calculating the projection between two vectors. But the vector or cross product is used for the two perpendicular vectors calculation. Now we can start our deep discussion about the cross or the dot product, algebraic operations, applications, and examples.
In the late 18th century, the Quaternion algebraic operation and the first products of the vectors which are violets the commutative law can be described by the scientist William Rowan Hamilton. The experiment can be performed by William in which he can do the product of two vectors and these are the quaternions and the other part for the product is zero which is scalar then their results also contain the vector or the scalar part. The part of scalar and the vector in the result of William product expressed the cross product of the two vectors which can be represented by the A B and the dot product of the two vectors can be expressed as the A B.
After this, the scientist Josiah Willard Gibbs in the 18th century 1881, with Oliver Heaviside represented the expression that can be used for the dot products of the two vectors and also for the cross product of the two vectors which are given there:
The dot product can be expressed through and written as;
A B
The cross product of the two vectors can be expressed through the cross and written as:
A B
As we can see the expression for both the dot and the cross product expressed that the vector A can be multiplied by the vector B and they can’t violet the commutative law so that's why their matrix can be always 3 3 and it can also be explained by the scientist, Saru's and their law or rule can be termed as Sarrus rule which is given there:
Cross or the vector product can be defined as:
“when the product of the two vectors is the vector quantity it can be represented as A B then it is teremed as the vector product or also the cross product. And the resultant vector which can be denoted by the C are perpendicular to the both of the vector A and the vector B.”
Mathematically the cross or the vector product can be written as:
A B = AB sinθ n
There,
A represented the vector A
B represented the vector B
And,
A represents the magnitude of the vector A
B represented the magnitude of the vector B
The θ represented the angle between the vector A and the vector B which lies in the 0° to 180°. And the unit vector which is perpendicular to the vector A and the vector B can be denoted through n.
The product of the two vectors, vector A and vector B is zero (0) when both of these vectors A and vector B are parallel to each other.
The magnitude and the direction of the vectors can be represented through the right-hand rule. In which the direction can be shown in the right-hand rule and the magnitude of two vector products is always equal to the parallelogram which is given or in which the vector product can be done.
The right-hand rule in the term of the cross or the vector product can be defined as:
"The thumb of the right hand determines the direction of the resultant vector C which is the product of two vectors cross product and when we can curl our finger in the direction of the thumb it indicates the direction the vector A and after proper curling of fingers, it indicates the direction of the vector B."
As we discuss the right-hand rule in terms of the cross or dot product the thumb and the curling finger represent the directions of the vector and also the direction of the resultant vector through the thumb.
In the given figure the cross or vector product of two vectors can be shown. The thumb represents the resultant vector which is equal to the product of two vectors A and the vector B
The fingers and the curl fingers can represent the direction of both vector's magnitude and the θ represents the angle between both of these vectors in the area of a parallelogram.
The product of the two vectors with their units vector, coordinate equation, or the mathematical expression are given there:
Let's suppose the two vectors, the vector A and the vector B which is equal to the,
A = A1 i + A2j + A3k
B = B1 i + B2j + B3k
As we know i , j and k are the unit vectors that can be oriented in the orthonormal basis positively and they can be written as:
k i = j
j k = i
i j = k
Now according to the anti-commutativity law, or when these unit vectors can be oriented negatively then the orthonormal basis can be written as:
i k = – j
j i = – k
k j = – i
Now when we can do the product of the two vectors, vector A and vector B with their unit vector which can follow the distributive law then it can be written as:
A = A1 i + A2j + A3k
B = B1 i + B2j + B3k
And
A B = (A1 i + A2j + A3k) (B1 i + B2j + B3k)
Then,
A B = A1 B1 ( i i ) + A1 B2 ( i j ) + A1 B3 ( i k ) + A2B1 ( j i ) + A2B2 ( j j ) + A2B3 ( j + k ) + A3B1 ( k i ) + A3B2 ( k j ) + A3B3 ( k k) …… (i) equation
We also know that:
i i = j j = k k = 0
Because the vectors are perpendicular and they can't follow the law of the commutative.
By putting the values of the unit vectors in the equation (i)
A B = A1 B1 (0) + A1 B2 (k) – A1 B3 ( j) – A2B1 ( k ) + A2B2(0) +A2B3 (i) + A3B1 ( j) – A3B2 (i) + A3B3 (0)
Then arrange them and then it can be written as;
A B = A2B3 (i) – A3B2 (i) – A1 B3 ( j) + A3B1 ( j) + A1 B2 (k ) – A2B1 ( k )
Now we take common the same unit vectors i, j, and k and write as,
A B = ( A2B3 – A3B2) i + ( A3B1 – A1 B3) j + ( A1 B2– A2B1 ) k
The products of the two vectors, vector A and vector B are the component of the scalar and their resultant vector are C which are equal and written as:
C = C1i + C2j + C3k
So that's why the resultant vectors with their unit vector are equal and written as:
C1i = A2B3 – A3B2
C2j = A3B1 – A1 B3
C3k = A1 B2– A2B1
Also, it can be written in the matrix, column matrix which is given there,
C1i C2j C3k |
A2B3 – A3B2 A3B1 – A1 B3 A1 B2– A2B1 |
=
To represent the vector products the determinants can be used and they can be written as:
i A1 B1 |
j A2 B2 |
k A3 B3 |
A B =
But if we can use the Sarrus rule in the matrix then it can be written as:
A B =( A2B3 (i) + A3B1 j + A1 B2k ) – ( A3B2 i + A1 B3 j + A2B1 k )
Then it can also be written as:
A B = ( A2B3 – A3B2) i + ( A3B1 – A1 B3) j + ( A1 B2– A2B1 ) k
And these are the components of the cross or the vector products.
The characteristics and the main properties of the cross or the scalar product are given there:
Area of a parallelogram
Perpendicular vectors
Self vector product
Violation of the commutative law
Parallel vectors
Anti parallel vectors
Vector product in the rectangular component
Distributivity
Scalar multiplication
Orthogonality
Zero vector
Their detail is given there:
The product of the two vector quantities, the magnitude of these vector A and vector B is equal to the area of a parallelogram along with their sides. The area of the parallelogram is equal to,
Area of a parllelogram= length height
Area of the parallelogram = ( A) (B sinθ)
there,
A represents the length
B represents the height
sinθ represents the angle between vector A and vector B
The total area of the parallelogram with their sides is the magnitude of these vector products. and it can be written as:
Area of a parallelogram = ( A B magniytude)
Also written as:
Area of a parallelogram = A B
When the two vectors, vector A and vector B are perpendicular to each other then their magnitude is always maximum because the angle θ between them is equal to 90°, then it can be written as:
A B = AB sin 90° n
As we know that:
sin 90° = 0
Then,
A B = AB (1) n
A B = AB n
And it is the maximum magnitude of the two vectors in the cross or vector product. But in the case of their unit vectors, it can also be written as:
k i = j
j k = i
i j = k
It can also be written as:
i k = – j
j i = – k
k j = – i
A A = AA sin 0° n
A A = 0° n
A A = 0 this is the zero or the null vector. Another example for the vector B is given there:
B B = BB sin 0° n
B B = 0° n
Then,
B B = 0
For the unit vectors, the self-product is also equal to the null or zero vector and written as:
i i = ii sin 0° n
i i = (1) (1) sin0° n
i i = 0° n
i i = 0
So the other unit vector self-product is also equal to the null or zero vector and can be written as:
j j = 0
k k = 0
And
i i = j j = k k
The cross or the vector product of the two vectors, vector A and the vector B are not to be commutative because they can't follow this law because the vectors are perpendicular to each other.
Let the vector A and the vector B and the sin θ be the angle that is present between the product of these two vectors and it can be written as:
A B = AB sinθ n ……… (i) equation
And if we can reverse them and write them as:
B A = BA sinθ(- n )
And also it can be written as;
B A = AB sinθ(- n ) ……… (ii) equation
According to the commutative law
AB = BA
Now compare the equation (i) and the (ii) equation
A B = – B A
A B ≠ – B A
Thus, it can proved that the cross or the vector product of the two vectors A and b can't follow the commutative property.
When the two vectors, vector, and vector B are parallel to each other then always their cross or the vector product is equal to the null or the zero vector.
Mathematical expressions for parallel vectors are given there:
A B = AB sinθ n
In parallel vectors θ = 0° then,
A B = AB sin (0) n
A B = AB (0) n
A B = (0) n
A B = 0
When the two vectors, vector, and vector B are anti-parallel to each other then always their cross or the vector product is equal to the null or the zero vector.
Mathematical expressions for parallel vectors are given there:
A B = AB sinθ n
In parallel vectors θ =180° then,
A B = AB sin (180) n
A B = AB (0) n
A B = (0) n
A B = 0
The cross or the vector product can be done in the cartesian or the rectangular components and there three components are the scalar but their product result is always the vector quantity.
Let's suppose the two vectors, the vector A and the vector B which is equal to the,
A = A1 i + A2j + A3k
B = B1 i + B2j + B3k
As we know i , j and k are the unit vectors that can be oriented in the orthonormal basis positively and they can be written as:
k i = j
j k = i
i j = k
Now according to the anti-commutativity law, or when these unit vectors can be oriented negatively then the orthonormal basis can be written as:
i k = – j
j i = – k
k j = – i
Now when we can do the product of the two vectors, vector A and vector B with their unit vector which can follow the distributive law then it can be written as:
A = A1 i + A2j + A3k
B = B1 i + B2j + B3k
And
A B = (A1 i + A2j + A3k) (B1 i + B2j + B3k)
Then,
A B = A1 B1 ( i i ) + A1 B2 ( i j ) + A1 B3 ( i k ) + A2B1 ( j i ) + A2B2 ( j j ) + A2B3 ( j + k ) + A3B1 ( k i ) + A3B2 ( k j ) + A3B3 ( k k) …… (i) equation
We also know that:
i i = j j = k k = 0
Because the vectors are perpendicular and they can't follow the law of the commutative.
By putting the values of the unit vectors in the equation (i)
A B = A1 B1 (0) + A1 B2 (k) – A1 B3 ( j) – A2B1 ( k ) + A2B2(0) +A2B3 (i) + A3B1 ( j) – A3B2 (i) + A3B3 (0)
Then arrange them and then it can be written as;
A B = A2B3 (i) – A3B2 (i) – A1 B3 ( j) + A3B1 ( j) + A1 B2 (k ) – A2B1 ( k )
Now we take common the same unit vectors i, j, and k and write as,
A B = ( A2B3 – A3B2) i + ( A3B1 – A1 B3) j + ( A1 B2– A2B1 ) k
The products of the two vectors, vector A and vector B are the component of the scalar and their resultant vector are C which are equal and written as:
C = C1i + C2j + C3k
So that's why the resultant vectors with their unit vector are equal and written as:
C1i = A2B3 – A3B2
C2j = A3B1 – A1 B3
C3k = A1 B2– A2B1
Also, it can be written in the matrix, column matrix which is given there,
C1i C2j C3k |
A2B3 – A3B2 A3B1 – A1 B3 A1 B2– A2B1 |
=
The formula which can be derived from the cross or the vector product can also be written in the form of a determinant and their mathematical expression is given there:
i A1 B1 |
j A2 B2 |
k A3 B3 |
A B =
The cross or the vector product of the two vectors follows the distributivity property. Their mathematical expression is given there:
A ( B + C ) = A B + A C
In cross or the vector product this distributivity property can be proved by the vectors.
The cross or the vector product of the two vectors A and vector B ( A B ) is always orthogonal to the vector A and the vector B.
The scalar multiplication can also be done with the cross or the vector product of the vectors. Their mathematical expression can be written there:
( cA) B = c ( A B)
c represented the scalar multiplication, A represents the vector A, and B represents the vector B.
In the various fields of science, vector or cross products can be used generally but in mathematics, computer graphics, physics, or engineering mostly cross or vector products can be used. Some applications of vector products with details are given there:
Computer graphics
Physics
Engineering
In computer graphics, wide cross or vector products can be used in different programs. The major parts in which the cross or the vector products are used are given there:
Rotations: in the graphics where the algorithm can be used the cross or the vector product is widely used. It can also be used to compute the angular velocity and also to determine the axis of the rotation. In animations or the different stimulating systems cross or the vector product can be used to simply them.
Normal vectors: for the lightening in the calculations in the computer graphing program the normal vectors are used generally. The non-parallel vectors which are lying in the programming of the vector or the cross product can be used to simplify them.
In physics, the cross or the vector product is widely used to solve complex algebraic operations along with geometry the main fields in which the vector product can be used are given there:
Angular momentum: angular momentum is the product of two different vector quantities, one is linear momentum which is denoted by ρ and the other is position vector which can be denoted by the r . Their formula or mathematical expression is given there:
L = p r
Their L denotes the angular momentum.
Angular momentum can be widely used in dynamic rotation or isolated systems.
Torque: torque is the product of two different vector quantities, one is force which is denoted by F and the other is position vector which can be denoted by the r . Their formula or mathematical expression is given there:
τ = F r
Their τ denotes the torque.
Application of vector products in engineering fields where mainly the cross product are used is given there:
Magnetic force: (B)
Moment of a force:F
With time or with the complexity of the quantities or algebraic operations cross or vector products can be used in many different new fields or they can also be improved the advanced topic mainly in which the cross or the dot product can be used are given there:
To simplify the complex vector problem or the complex problem in physics the triple product of the vectors can be used because it can simplify them in a very efficient or accurate way. The mathematical expression or the formula that can be used in vector triple product is given there:
A ( B C ) = ( A . C ) B – ( A . B) C
there,
A represented the vector A
B represented the vector B.
C represented the vector C.
Three vectors can be used in this product so that is why it can also be termed as the triple vector product.
Angular momentum: angular momentum is the product of two different vector quantities, one is linear momentum which is denoted by ρ and the other is position vector which can be denoted by the r . Their formula or mathematical expression is given there:
L = p r
Their L denotes the angular momentum.
Torque: torque is the product of two different vector quantities, one is force which is denoted by F and the other is position vector which can be denoted by the r . Their formula or mathematical expression is given there:
τ = F r
Their τ denotes the torque.
Force of a moving charge: force in the magnetic field that can apply on the charging particle is the product of the two vectors and they are the velocity of the charged particle and the other vector is the magnetic field. their mathematical expression is given there:
F = q ( v B)
There,
F denote the force of the charged particles in the magnetic field.
q denotes the charge of the particles
v denotes the velocity of the charged particles.
B denotes the magnetic field.
Like the scalar or the dot product the cross or the scalar product can play a very vital role in different fields of science and simplify complex quantities or solve complex algebraic problems in engineering, physics, and mainly in mathematics. Cross or the vector product can show the expressing relation between the algebra and the geometric calculations and solve the problems in a very efficient way in Euclidean space. After understanding the applications and the depth of the cross or the vector product, easily the complex problem can be simplified efficiently. With time the cross or vector product is more commonly used in various fields of science.
Hi friends, I hope you are all well. In this article, we can discuss the scalar or dot products of the vectors. In previous articles, we have discussed vectors and their addition in the rectangular or cartesian coordinate system in depth. Now we can talk about the scalar product of two vectors, also known as the dot product. Scalar or dot products can play an essential role in solving the operation of vector algebra and also they have various applications in numerous fields like computer sciences, mathematics, engineering, and physics.
By doing the scalar or dot products, two vectors are combined when we can do their product, then they produce the scalar quantity which has both magnitude and direction by a single operation in a very efficient way. Simply the scalar and the dot product are algebraic operations that can be especially used in physics and mathematics. scalar quantity can only provide the magnitude but when we can do the product of two vectors, the result of this product is scalar quantity which provides and describes both magnitude and direction. The angle between the two vectors can also be found through the scalar or dot product. The dot product term can be derived from the word dot operator and it can be used for the product of two vectors but it can also known as a scalar product because it can always give the result as a scalar quantity so that is why it can also be known as scalar product rather than the vector product.
Now we can start our detailed discussion but the dot or the scalar product, their definition, algebraic operations, characteristics, applications, and examples. At the end of this discussion, the reader easily understands vectors, how we can make the scalar product, and their application in numerous fields of science, especially in physics or mathematics.
Dot/Scalar products can be defined geometrically or algebraically. But in the modern form, the scalar and the dot product can be defined and rely on the Euclidean space which has the cartesian or rectangular coordinate system. The basic and simple definition of the scalar and the dot product are given there:
“The product of two vectors is a scalar quantity so that's why the product is termed scalar product”.
The mathematic expression which can express the dot or scalar product is given there:
A B = AB cosθ
Where,
A is the magnitude of the vector A.
B is the magnitude of the vector B.
And,
The cosθ is the angle between the two vectors A and the vector B.
The dot product or the scalar product produces a single scalar quantity which can be produced through their mathematical operation. The product of the two vectors based on orthonormal base or in n-dimensional space, their mathematical expression or definition are given there:
A B = A1b1 + A2B2+ ……… + AnBn
There;
A = A1 , A2, ........ , An
B = B1, B2 , ......... , Bn
A B can also be mathematically written as;
A B = i=1naibi
there n represented the dimension of the vector in the Euclidian space or the summation is represented through.
For example, the dot or scalar product of the vector A = ( 5, 4, 4) or vector B = (2, 1, 6) in the three dimensions is calculated as:
A B = A1b1 + A2B2+ ……… + AnBn
By putting the values we can get,
A B = ( 5 2) + ( 4 1) + ( 4 6)
A B = 10 + 4 + 24
A B = 38
moreover, the vectors ( 6, 3, -2 ) themselves can do dot or scalar products which can be written as:
( 6, 3, -2) ( 6, 3, -2) = (6 6) + (3 3) + (-2 -2)
= 36 + 9 + 4
= 49
Another example for the dot or scalar product of the vector A= ( 4,6) and the vector B= ( 2, 8) in the two dimensions can be expressed or calculated as:
( 4, 6) ( 2,8 ) = ( 4 2) + ( 6 8)
= 8 + 48
= 56
The product of the two vectors can also be written in the form of a matrix. The formula that can be used for the matrix product of two vectors can be written as
A B = At. B
There,
At = transpose of the vector A
For instance,
4 3 2 4 9 4 then this matrix has vectors column 1 1 = 1
And the column in this vector is 3 3 = 6
In this way, we can write the vectors in the matrix row or column form and the result is a single entity.
In geometry, Euclidean vectors can describe both magnitude and direction through the scalar product or from the dot product. The length of the vector represents the magnitude and the direction of these vectors can be represented through the arrow points that are present on the vectors. The scalar and the dot product in geometry can be written as;
A B = A B cosθ
There,
A represented the magnitude of the vector A.
B represented the magnitude of the vector B.
And,
θ represented the angle between the magnitude of the vector A and the vector B.
If the vector A and the vector B are orthogonal then the angle between them θ = 90° or also equal to the π2 it can be written as:
A B = cosπ2
hence,
The cosπ2 is equal to 0. It can be written as:
A B = 0
If the vector A and the vector B are codirectional then the angle between their magnitude is equal to 0. Then,
A B = cos 0
hence,
cos0 = 1 and written as:
A B = A B
If the vector A does scalar or the dot product itself then it can be written as:
A A = A2
That can also written as:
A = A . A
This formula can be used to determine the length of the Euclidean vector.
The simple physical meaning of the scalar or dot product is that the product of the dot or scalar product is equal to the magnitude of the one vector and the other is equal to the component of the second vector which is placed in the direction of the first vector.
Mathematically it can be expressed as:
A B = A ( projection of the vector B on the A).
A B = B (the component of vector B magnitude along with the vector A )
Then it can also be written as:
A B = A ( B cosθ )
Then for the vector B we can write as:
B . A = B ( projection of the vector A on the vector B)
B . A = B ( the component of vector A magnitude along with the vector B).
Then it can also be written as:
B . A = B ( A cosθ)
The other physical meaning or the projection of vectors with their first property can discussed in detail. the projection of vector A in the direction of the vector B can also be written as:
Ab = A cosθ
The θ is the angle between the two vectors A and the vector B.
This product can also be written according to the definition of geometrical dot product then it can be written as:
Ab = A B
There,
B = BB
so, geometrically we can write the projection of A on the vector B as:
A B = Ab B
For the vector B, it can be written as:
A B = BaA
The dot product can also prove the distributive law, the distributive law is written as:
A ( B + C) = A B + A C
This law can be satisfied by the dot product because the scaling of any variable is homogenous. For example, if we can take the scalar B then it can be written as:
( BB) A = B ( B A)
Also written as,
( BB) A = B ( B A )
The dot product of the B B is always positive it never be negative but it may also equal to zero.
Determine the standard basic vectors E1, E2, E3, ……., En. So we can also write this as:
A = A1, A2, A3, ...... , An also equal to iAiEi
B = B1, B2, B3, ...... , Bn also equal to iBiEi
This formula Ei can represent the unit length of the vectors. Also represented that the length of the unit is at the right angle.
The normal unit length of the vector is equal to 1 and written as:
Ei Ei = 1
But when the length of unit vectors is at the right angle then it can be written as:
Ei Ej = 0
there, i ≠ j.
Basically, we can write the all formulas as:
Ei Ej = δij
there, Ei or the Ej represented the orthogonal vectors unit length and the δij represented the Korenckar delta.
According to the geometrical definition of the dot or scalar product, we can write the given expression for any different type of vector A and the vector Ei. the mathematical expression is written as:
A Ei = A Ei cosθi
or,
Ai = A cosθi
Now apply the distributive law on the given formula which is according to the geometrical scalar product or the dot product. The distributive version of this formula is given there:
A B = A i BiEi
It can also equal to,
= i Bi( A Ei)
= i Bi Ai
= i AiBi
Now it interchangeability of all definitions can be proved. It can be shown that all definitions of formulas are equal to each other.
In the dot product or the scalar product the geometrical interpretations are essential because they can relate the magnitude of the vectors through the dot product and the dot product can also give the angle between the vectors which are cosine. The main geometrical interruptions are given there:
Projection
Orthonogolity
Parallel vectors
Anti-parallel vectors
Their details are given there:
By the dot or the scalar products, we can measure the direction and the projection of the vector how much the vector lies on the other vector in the projected direction. For instance, A B through we can measure the projection of vector A on the vector B in a very efficient way.
When the two vectors are perpendicular to each other, then their dot or the cross product is zero because the angle θ is equal to 90 degrees and the cos90 degree is equal to zero. So if the dot or cross product of the vector quantity is zero then it means that the vectors are orthogonal.
In the dot or the cross product, if the vectors are parallel then the angle θ is equal to 0 degrees then the cos θ = 1, which means that the dot or cross product reached their maximum positive or negative magnitude.
In the dot or the cross product, if the vectors are anti-parallel then the angle θ is equal to 180 degrees then the cos θ = 1, which means that the dot or cross product reached their maximum positive or negative magnitude.
The main properties and the characteristics of the scalar and the dot product which can help to understand the dot or scalar product are given there. By understanding pr follow the given properties we can easily use this dot product in different fields of science and physics. The characteristics in detail are given there:
Distributive property
Parallel vectors
Anti parallel vectors
Self scalar products
Scalar multiplications
Commutative property
Perpendicular vectors
Magnitude
Product rule
Orthogonal
Scalar product in the term of rectangular component.
Zero vector
The distributive property of the dot or the scalar product can be strewed upon the vector addition. The basic and general expression for the distributive property for the dot or cross product is given there:
A ( B + C ) = A B + A C
The scalar or dot product of the two vectors is equal to their positive magnitude when the vectors which are used in the dot or scalar product are parallel to each other and their angle θ is equal to 0 degrees, it can be written as:
θ = 0°
The mathematical expression for parallel; vector can be written as:
A B = AB cos 0°
and, cos 0° equal to 1 and written as:
A B = AB (1)
A B = Ab
hence, it can be shown in the above equation that the product of two vectors is equal to their magnitudes. It can also be the positive maximum value of the scalar or the dot product.
The scalar or dot product of the two vectors is equal to their negative magnitude when the vectors which are used in the dot or scalar product are anti-parallel to each other and their angle θ is equal to 180 degrees, it can be written as:
θ = 180°
The mathematical expression for an anti-parallel vector can be written as:
A B = AB cos 180°
and, cos 0° equal to 1 and written as:
A B = AB (-1)
A B = -Ab
hence, it can be shown in the above equation that the product of two vectors is equal to their magnitudes. It can also be the negative maximum value of the scalar or the dot product.
The dot product or the scalar product can directly affect the scaling of the vector. Through this property of the dot or cross product, we can observe this effect efficiently. The equation that can be used for the scalar multiplication property is given there:
(c1 A ) ( c2 B ) = c1c2 ( A B)
There c represented the scalar quantity.
When the vector can do their self-product then the result is always equal to the square of their magnitudes.
The basic and the general equation is written below:
A B = AA cos 0°
A B = AA (1)
A B = A2
It can be shown in the given equation that the self-product was always equal to the square of their magnitudes.
Self product of unit vectors:
The self-product of the unit vectors is always equal to the 1. Their clarification through mathematical expression is given there:
i i = (1) (1) cos 0°
i i = (1) (1) (1)
i i = 1
So,
j j = 1
k k = 1
Hence,
i i = j j = k k
The scalar or the dot product of two vectors A and B are always commutative. Their mathematical justification is given there:
A B = AB cos θ …….. (i) equation
there, A represented the vector
B also represented the other vector
And θ represented the angle between the vectors A and B.
then,
B A = BA cos θ ………. (ii) equation
Then, by comparing the equation i and the equation ii,
A B = B A
Hence proved that the dot or scalar product is always commutative.
In the product of two vectors if one vector A = 0 then the other vector B = 4 but their product is always equal to zero. Their mathematical expression is written there as:
= A B
= (0) (4)
Then,
A B = 0
If the two vector scalar or dot products are equal to zero then it can't be orthogonal but if the two vectors are non-zero variables it can be orthogonal.
In the scalar or the dot product, the values are different or variable and their deviation can be represented through the sign which is known as the prime ′. Their mathematical expressions are given there:
( A B) ′ = A′ B + A B′
Determine the two vectors, the vector A and the B in the Euclidean space in the three-dimensional cartesian coordinate system. Their derivation is given there:
Let,
A = Axi + Ayj + Azk
B = Bxi + Byj + Bzk
then, we can perform their product with their unit vectors and it can be written as:
A B = (Axi + Ayj + Azk) (Bxi +B yj + Bzk)
After this, we can multiply the all components with each other and it can be written as:
A B = AxBx ( i i ) + AxBy ( i j) + AxBz ( i k) + AyBx ( ji )
AyBy ( j j ) + AyBz ( jk ) + AzBx ( ki) + Az By ( k j) + Az Bz( k k)
Now, by putting the values of the unit vectors then we get,
A B = AxBx (1 ) + AxBy ( 0 ) + AxBz ( 0 ) + AyBx (0 )
AyBy (1 ) + AyBz ( 0 ) + AzBx ( 0 ) + Az By (0) + Az Bz(1)
Then,
A B = AxBx + AyBy + Az Bz ……… (i) equation
We know that ;
A B = AB cos θ ………… (ii) equation
Then put equation (ii) in equation (i) and we get,
AB cos θ = AxBx + AyBy + Az Bz
Or it can also be written as;
cos θ = AxBx + AyBy + Az BzAB
Or,
θ = cos-1AxBx + AyBy + Az BzAB
This formula or the equation can be used to find the angle θ between the vector A and the vector B.
Scalar or dot products can play a very essential and fundamental role in different fields of modern science or physics, computer graphics, engineering, or data analysis. The details of these applications are given below:
Data analysis or machine learning
Mathematics
Physics
Engineering
Computer graphics
Dot or scalar products can be used in data analysis or machine learning in a very efficient way their applications in this field mostly occur in the given fields area which are;
Natural languaging processing
Principal component analysis
Neural networks
Their description is given below:
The differences and the similarities that may be present in the natural languaging processor ( NLP) can be detected through the scalar and the dot product because the words can be represented in the form of vectors in NLP. and it can also help to do many numerous tasks like the machine translation, data analysis and the document clustering in a very efficient way.
Principal component analysis which can also be denoted as PCA, to determine and find the principal components that are present in the data can be detected by using the dot or cross-product method. Because it can simplify the most complex data or analyze them in a very efficient way. So that's why cross or dot products can be widely used in this field.
For the sum of the neurons, we can use the dot or the scalar product. Because of all the neurons, the input vector calculation can always be done through the dot or the cross product, and by the activation the output can be produced.
In mathematics, the dot and cross product can be used commonly because geometry and the algebraic operation can be solved easily or efficiently through the dot and the cross product. The main fields of math in which the dot and cross product can be used are given there:
Cosine similarity
Orthogonality
Projection
Vector spaces
In physics to simplify the complex quantities and products dots or scalar products can be used. The main application fields are given there:
Molecular dynamics
Work done
Electromagnetic theory
In engineering, algebraic operations can be simplified efficiently through the dot or scalar product. But the main areas of this field where mainly dot and scalar products can be used are given there:
Robotics
Signal processing
Structural processing
Like other fields of science, the dot and the scalar product can also be used in computer graphics because through using the dot or scalar product we can efficiently understand or solve the complex codes of words that can be represented in the form of vectors.
Vector projection
Lighting calculations
Shading models
Work done:
Work is the scalar quantity but it can be a product of two vector quantities through the dot or scalar product. The product of force and displacement produced the scalar product work. Which can be written as:
W = F A
There,
F represented the force.
A represented the displacement.
Calculation:
Consider the force F is ( 4, 5) and the displacement of the object is ( 2, 8 )
Then their product can be written as:
W = F A
By putting the values we can get,
W = ( 4 5) ( 2 8)
W = (20) (16)
W = 36 J
The work that can be done by the body is equal to the 36 J.
Magnetic flux:
The magnetic flux is the product of the two vectors which are magnetic field strength and the vector area which can be expressed as:
Øb = B A
Power :
Power( scalar product ) is the product of two scalar quantity which are force and velocity which are expressed as:
P = F v
Electric flux:
Flux is the scalar quantity and it is the product of the two vector quantities which are electric intensity or the vector area. it can be written as:
Øe = E A
Hi friends, I hope you are all well and doing good in your fields. Today we can discuss the vector quantities and how we can add the vector by rectangular components. Generally, there are two quantities one is scalar quantities and the other is vector quantity. Scalar quantities are those quantities that have only magnitude but vector quantities are those that can describe both magnitude and direction. So in physics or for complex quantities vectors are used because they can describe both magnitudes with direction.
Vectors can play a very fundamental role in the different fields of physics and mathematics because they can provide accurate and precise measurements. In rectangular components, we can add two or more vectors by breaking them according to their planes. The most efficient method for adding the vectors is adding vectors in rectangular components. Now in this article, we can start our detailed discussion about the vectors and their addition by the method of the rectangular component.
Vectors can be defined as quantities that can describe both magnitude and direction but they can't provide a description about the position of a quantity. Vectors can be used to describe complex physical quantities like velocity, displacement, and acceleration. Vectors can also used to express the mathematical form of laws and in geometry firstly vectors are used. Some more examples of the vector quantities are given there.
Vectors which may be two or more two can be added by rectangular component because they are the cartesian coordinate system. now the main point about what are rectangular components and their mathematical expression are given there.
In the graph or two-dimensional cartesian coordinate plane, there are axsis which are usually x and y these axsis are known as rectangular components for vectors. But if the cartesian coordinate plane is three-dimensional then the three planes and components are x, y, and z.
For example, if we have the vector A then their components on the two-dimensional cartesian plane are Ax and Ay. But if we have the vector B on the three-dimensional plane then their rectangular components are Bx, By and Bz
A: represent vector A
Ax : represent the component of a vector A along with the x-axis
Ay : represent the component of a vector A along with the y-axis
And if they are three-dimensional then,
Az: it can represent the vector A along with the z-axis in the three-dimensional cartesian plane.
i, j and k : these are the unit vectors that can be used according to their rectangular components like i the unit vector of x- the x-axis rectangular component, j the unit vector of the y-axis of the rectangular component, and the unit vector k for the z-axis.
Now we know about the rectangular components but if we want to add the vectors by using the rectangular component first we can decompose the vectors according to their component.
In a dimensional cartesian plane, there are two components x and y so that is why the vector A has the magnitude A and also has the angle 𝚹 on the x-axis. Their decomposition equation is given there:
A = Axi + Ayj
Where,
Axi: A cos𝚹
Ayj : A sin𝚹
In three three-dimensional cartesian planes the x, y, and z are the components for the vector A then there decomposition of rectangular components can be written as:
A = Axi + Ayj + Azk
Vector addition by rectangular component is also known as the Analytic method of vector addition. This method can add the vectors efficiently and the chances of error are very low as compared to other methods like the head-to-tail rule or other graphical methods. Now we can start the vector addition by rectangular components.
Let's imagine we have two vectors one vector A or the other is a vector B now we can add them to the rectangular cartesian coordinate system and suppose their resultant is R and these vectors make an angle θ on the x-axis. By using the head-to-tail rule the resultant of two vectors which is A or B are R = A + B now we can resolve the vectors A, B and the resultant vector R into their rectangular components.
Now in this figure, the vector addition is shown and the rectangular components of the vector A, B and the resultant vector R are also shown now we can start our derivation to resolve the all vectors in the figure.
Firstly we can find the x component of the resultant and the y component of the resultant.
As shown in the figure,
And,
then,
Then according to the given figure, we can write these magnitudes of the vector as:
OR = OQ + QR
Since the QR is also equal to the MS. we can write it as,
OR = OQ + MS
And according to the vectors it can be written as:
Rx = Ax + Bx ……………. (i) equation
The sum of the magnitude of the vector A and the vector B on the x component is equal to the magnitude of the resultant vector R on the x component which can be shown in the (i)equation.
As shown in the figure,
and,
then,
RP is the magnitude of the resultant vector R on the y component which is shown in the figure.
Then according to the given figure, we can write these magnitudes of the vector as:
RP = RS + SP
According to the given figure the RS is also equal to the QM so we can also write the equation as;
RP = QM + SP
Now this equation can be written according to the vectors as:
Ry = Ay + By ………… (ii) equation
The sum of the magnitude of the vector A and the vector B on the y component is equal to the magnitude of the resultant vector R on the y component which can be shown in the (ii)equation.
Now we can write the resultant vector on the x component or y component with their unit vectors.
The resultant vector of the x component with its unit vector is written as Rx i.
The resultant vector of the y component with its unit vector is written as Ryj.
Then the resultant vector with its unit vector in the equation can be written as:
R = Rxi + Ryj
Now we can put the values of Rxi or Ryj in the resultant vector R.
R = Rxi + Ryj
Putting the values from the equation (i) and equation (ii) and written as
R = ( Ax + Bx) i + ( Ay + By ) j
This equation is used to add the vectors on the rectangular components.
After adding the vectors on the rectangular component we can also find their magnitude by using some formula. The formula which we can use to find the magnitude of the resultant
R is given there:
R = Rx2 + Ry2
And if we want to find the magnitude of the vector A and vector B we can put the values of the resultant vector Rx and the resultant vector Ry in the given formula and we can write this formula as:
R = (Ax+Bx )2+ (Ay+By)2
This formula can be used to find the magnitude of the vectors that can be added to the rectangular component.
But if we can find the magnitude of the resultant R which has the vectors A and vector B then we can also use this formula which is given there :
R = A2 + B2 + 2ABcosӨ
There are some special cases in which the value of θ can be different so we can change some formulas. Some special cases are given there:
If the value of θ = 90° then,
R = A2 + B2
But if the value of θ= 0° then,
Rmax = A + B
And if the value of θ=180° then,
Rmax = A – B
Vectors can describe the magnitude but they can also describe the direction so after finding the magnitude we can also find their direction by using the formula. To find the direction of the resultant vector R we can use the formula which is given below:
tanθ = RyRx
Also, it can be written as:
θ = tan-1 RyRx
But if we want to find the direction of the vectors A and B we can put the values of Rx and Ry. and it can be written as:
θ = tan-1 AY+ByAx+Bx
These all formulas can be used for two-dimensional vectors but if we want to find the three-dimensional vector or many other vectors we can use the other formulas that are given there.
The two vectors A and the vector B can lie in the three dimensions in the rectangular cartesian coordinate system.
The components of the resultant vectors in three dimensions are given there:
Rx components on the x-axis: Ax, Bx
Ry components on the y-axis: Ay, By
Rz components on the z-axis: Az, Bz
The components of vectors A and B in the three dimensions are given there:
A = Axi + Ayj + Azk
B= Bxi + Byj + Bzk
A sum of these two vectors in the three dimensions is given there:
R = Rxi + Ryj + Rzk
Then put the values and get the equation which is given there:
R = (Ax+Bx) i + (Ay+ By) j + (Az+ Bz) k
This formula is used for the two vectors on the three dimensions.
We can also add the multiple vectors in the two dimensions. Then the resultant components on the x, y, and z axes with their vector components are given there:
For the vectors A1, A2 and the vector An.
then,
R = i=1n Ai
Rx = i=1nAix
Ry= i=1nAiy
The formula that can be used for resultant vectors in these three dimensions is given there:
R = Rx2 + Ry2 + Rz2
To find the magnitude of the coplanar vectors A, B, C, D and ........ we can use the formula which is given there:
R = (Ax+Bx+Cx +...........)2+ (Ay+By+Cy+..........)2
To find the direction of the coplanar vector we can use this formula which is given there:
θ = tan-1Ay+By+Cy..........Ay+BY+Cy+...........
By using the given formula we can first determine and find the θ.
θ= tan-1RyRx
After the determination of the angle check the signs of Rx and the Ry in the rectangular cartesian coordinate system and determine their resultant quadrant according to their signs.
Determine the resultant quadrant through the signs of Rx and the Ry. The rules which can be followed to determine their quadrants are given there:
The resultant vector R lies in the first quadrant if the sign is positive for both of them Rxand the Ry vectors. Their direction is
θ = Φ
The resultant vector R lies in the second quadrant if the Rx is negative and the other vector Ry is positive. And their direction is,
θ = 180° – Φ
The resultant vector R lies in the third quadrant if the Rx and the Ry both are negative no one from them is positive. Their direct is,
θ = 180° + Φ
The resultant vector R lies in the fourth quadrant if the Rx is positive and the other resultant vector Ry is negative. Their direction is,
θ = 360° – Φ
For adding the vectors in the rectangular components in a very efficient way we can use some rules. These rules are as given below:
Vectors: First we can determine the x and y components for all vectors in two dimensions and if they are three-dimensional addition then also find the z components of all vectors.
Resultant vector Rx: then to find the resultant vector Rx which is the x component, add all the vector components on the x axes.
Resultant vector Ry: then to find the resultant vector Ry which is the y component, add all the vector components on the y axes.
Magnitude: After this, we can find the magnitude of resultant vectors by using the given formula:
R = Rx2 + Ry2
Direction: then we can also find the direction of the vector along with the magnitude by using the given formula:
θ= tan-1RyRx
Some examples of adding the vector in a 2D or 3D rectangular system are given there:
The given vectors are A and vector B.
The values of these vectors are:
A= 6i + 4j
B = -4i + 3j
Then add by using the resultant formula R = A + B
Then find Rxand Ry and add them,
Rx= 6 + (- 4) = 2
Ry = 4 + 3 = 7
Then,
R = 2i + 7j
Magnitude:
R= Rx2+ Ry2
By putting values,
R = 22+ 72
R= 4+ 49
R= 53
R≈ 7.280
After finding the magnitude we can find direction by using the given formula:
θ = tan-1RyRx
By putting the values we get,
θ = tan-172
θ ≈ 16.35
The given vectors are A vector B and the vector C.
The values of these vectors are given there;
A= 6i + 4j + 1k
B = -4i + 3j + 5k
C = -1i + 3j + 2k
Then add by using the resultant formula R = A + B + C
Then find Rxand Ry and Rz and then add them,
Rx=6+ (- 4) +(-1) = 1
Ry=4 + 3 + 3 = 10
Rz= 1 + 5 +2 =8
Then,
R = 1i + 10j +8k
Magnitude:
R= Rx2+ Ry2+Rz2
By putting values we get,
R = 12+ 102 + 82
R= 1 + 100+ 64
R= 165
R ≈ 12.85
The addition of the vectors in the rectangular components can be used in different fields of physics because it is an analytic method and provides precise and accurate calculations so scientists in physics or mathematics use this method for the calculation of complex physical quantities. Now we can discuss some applications of adding vectors by rectangular components in some different fields.
To determine for find the orientation and position of the robot's arms or legs in an efficient way we can use the vector addition or analytic method because it can provide accurate information without any possible errors.to achieve coordination and control of the robots we can also use the vector addition method by decomposing their component according to the axis.
For the analysis of the vector quantities like velocity, displacement, acceleration, and force in the accurate or precise method we can use the analytic or the vector addition by rectangular component method. In navigation, if we want to calculate the resultant velocity we can use this analytic method by adding the vectors which are the velocity of the wind and the velocity of the aircraft from which they can fly. This, there are many examples in which this method can be used for calculating the quantities. For example, many external forces can act on the body then we can all add them by using this analytic method and get the sum of all external forces which can act on them.
In the field of computer graphics, we can transform the objects their position, and movements, and we can calculate all of these movements through vector addition or analytic methods. The complex motion of the objects their movement, position, and all control on them was handled efficiently through breaking down their components according to their rectangular components axes.
With time or in the modern era of science and technology vector addition can be used in many new different fields some are given there:
The vector A in the spherical coordinates their magnitude can be represented through the A and the angle between them is represented through θ and also represented through the azimuthal angle Φ. In spherical coordinates the vector addition or analytic method can also be used to decompose the components, adding them or also convert them into their original form.
Vectors can be added basically or generally into the rectangular or cartesian coordinate system but vectors can also be added in many different coordinate systems like polar, polygram, cylindrical, or in many different spherical coordinate planes. But in different spherical coordinate systems, we can follow many other different rules which may be addition or different from the addition of vectors in the rectangular coordinate system.
The vector A in the polar coordinate system, their magnitude can be represented through A and the angle can be expressed through θ. But the addition in the polar coordinate system is difficult so that's why if we want to add the vector in the polar coordinate system we can convert them, decompose them, and then add them into the rectangular component and if the need arises we can convert them and then added them.
In vector addition in its rectangular components, some mistakes can occur when the vectors are complex and the calculation becomes challenging. Some common mistakes and challenges are given there:
In the vector addition or during the calculations units can play an essential role but if we can neglect them and can't track them properly then the inaccurate calculation or result from chances increases if we can track the units properly then there is no chance for error and the result are accurate and efficient. Mixing up of units can also provide inaccurate or false results.
When we add these vectors to the cartesian or any coordinate system it is essential to check their coordinates and components properly because if any vector lies on the wrong coordinate plane the result is incorrect. Coordinate planes can play a very essential role in a vector addition misleading coordinate axes always provide inaccurate calculations and results.
When we can perform the trigonometric functions the chances of error are possible but if we can check the calculations again and again then there is no chance of error. If the signs and values of vectors according to their components are not correct then their calculation results are also inaccurate. Because they can cause different significant errors so that's why double double-checking the units and the components in the coordinate plane is essential for precise and efficient results.
In different fields of physics or mathematics or many others like engineering, robotics, and computer graphics vector addition can play a very essential and powerful role also vector addition can be handled and provide control on different types of robots. Vector addition can also play an essential role in understanding complex vector quantities and also help to understand the theory of trigonometrics and resolve complex trigonometric problems in a very efficient way.
The critical value serves as a boundary that defines a specific range where the test statistic acquired during hypothesis testing, is improbable to lie within. The critical value is a benchmark against which the obtained test statistic is compared during hypothesis testing. This comparison helps in deciding whether to reject the null hypothesis or not.
On a graph, the critical value explains the boundary between the acceptance and rejection areas in hypothesis testing. It aids in judging the statistical importance or significance of a test statistic. This article will explain the following basics of critical value:
What is the critical value?
Types of Critical Value.
How to Calculate Critical Value?
Examples of Critical Value.
Summary.
A critical value is a specific test statistic value that establishes a confidence interval’s limits (upper and lower). It also sets the threshold for determining statistical significance in a statistical test. It indicates the distance away from the average (mean) of the distribution needed to comprehend a particular portion of the overall variation in the data (Such as 90%, 95%, or 99%).
There are various types of critical values used in statistical analysis which depend on the nature of the test and the distribution being considered. Some of the common types include:
Z-Score Critical Values
T-Score Critical Values
χ² (Chi-Square) Critical Values
F-Statistic Critical Values
Used in hypothesis testing for population means when the population standard deviation is known. Z-scores correspond to specific percentiles of the standard normal distribution.
Specify the alpha level.
Compute 1 minus the α level to derive the adjusted value for a 2-tailed test. Deduct the alpha level from 0.5 in the case of a 1-tailed test.
Use the z distribution table to find the area and obtain the z critical value. In the case of a left-tailed test, include a negative sign to the critical value after obtaining it.
Used in hypothesis testing for population means when we don't know the population standard deviation and have a small sample size. T-scores come from the t-distribution and vary based on the degrees of freedom.
Used in chi-square tests for independence or goodness of fit. Chi-square critical values are based on the chi-square distribution and depend on the degrees of freedom and the chosen level of significance.
Calculate the degrees of freedom for the Chi-Square distribution.
Choose the significance level (α).
Refer to the Chi-Square distribution table.
Locate the critical value corresponding to the degrees of freedom and chosen significance level.
Utilized in ANOVA (Analysis of Variance) tests to compare variances between multiple groups. F-statistic critical values are taken from the F-distribution and depend on the degrees of freedom of the groups being compared.
Compute the alpha level.
Deduct one from the size of the initial sample to get the degree of freedom. Denoted as X.
Deduct one from the second sample to determine the 2nd Degree (df) of freedom. Labeled as y.
Consult the F distribution table, locating the value where the column representing x intersects with the row representing y. This intersection provides the F critical value necessary for the analysis.
To calcauate the different critical values first we need to select the test and use the related distribution table according to the test. There’s no universal formula and methods for finding the value of the critical values, it just depend on the test selection. Follow the below steps to find the critical value using different test:
Type of hypothesis test: Z-test, t-test, chi-square test, F-test, etc.
Level of significance (α): Typically, 0.05 or 0.01.
Tailed Ness of the test: One-tailed or two-tailed.
Degrees of freedom (df): Often needed for t-tests, chi-square tests, and F-tests.
The critical values for both 1-tailed and 2-tailed tests can be determined using the confidence interval. The process to calculate the critical value is as follows:
Deduct the confidence level from 100%.
Change this value into decimal form to obtain α (alpha).
If the test is 1-tailed, the alpha level remains the same as in step 2. However, for a 2-tailed test, the α level is divided by 2.
The critical value can be determined by consulting the relevant distribution table, based on the type of test and the alpha value.
Examples Related to Critical Value
In this section, we calculate the different critical values using the its respective test and formulas. For the better underestnfding of the calculations provide the detailed steps.
Example 1: Chi-Square Critical Value Calculation
Suppose you’re conducting a chi-square test to analyze the independence between two categorical variables in a survey. Your contingency table has 3 rows and 4 columns.
Solution:
Calculate Degrees of Freedom (df):
For a chi-square test of independence, degrees of freedom (df) are computed as:
df = (Number of rows - 1) * (Number of columns - 1)
df = (3 - 1) * (4 - 1)
df = 2 * 3 = 6
Choose Significance Level (α):
Suppose we are working with a significance level at α = 0.05, which is commonly used in hypothesis testing.
Refer to Chi-Square Distribution Table:
Consult the Chi-Square distribution table with 6 degrees of freedom and α = 0.05.
Locate Critical Value:
Find the critical value for 6 degrees of freedom at α = 0.05 in the Chi-Square distribution table.
Chi-square Distribution Table
Critical Value = χ² = 12.592 (df=6, α=0.05)
Alternatively, you can use the Critical Value Calculator to determine the critical value quickly, saving your time and efforts by manual calculations.
Example 2: Z – score Critical Value Calculation
Suppose we're conducting a hypothesis test to determine if the average IQ of a population is significantly different from a claimed mean IQ of 100, with a population standard deviation known to be 15. We'll perform a two-tailed test at a significance level (α) of 0.21.
Solution:
Specify the alpha level: α = 0.21 (significance level)
Compute 1 minus the alpha level for a two-tailed test: For a two-tailed test, the adjusted alpha level is
1 - α = 1 - 0.21 = 0.79.
Use the Z-distribution table to find the critical Z-value:
Therefore, the critical Z-score for a two-tailed test is approximately ± 0.81.
This article explored the essential concept of critical values in hypothesis testing. We understood their role in defining boundaries for the test statistic and judging its statistical significance. We delved into the formulas and steps for calculating critical values for various scenarios like Z-tests, T-tests, and Chi-Square tests. We examined different types of critical values and observed their application in practical examples.