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.
Hi readers, I hope you are all well. In this post, we can talk about the vectors briefly. Physical quantities can be defined through magnitudes but some physical quantities can be defined through both magnitudes and direction, these types of quantities defined through both magnitude and directional properties are known as vectors, and the quantities that can be explained through magnitude, not with direction are known as scalars. some vectors are force, velocity, displacement, and acceleration.
Vectors can explain the direction and magnitude of the quantity but they can't provide their position. It is an essential tool of mathematics that can be used in physics for knowing the direction or magnitude. It cant be used in the 18th century but can be used in the modern era of the 19th century late and it can be presented by the scientists Oliver Heavisde and Willard Gibbs they can develop the vector analysis which can be used to express the modern laws of electromagnetism which can be presented by the James clerk maxwell.
In different fields of physics like mechanics, and mathematics or in engineering, vectors can be used to explain the different qnatites in mathematical form with magnitude and direction. Now we can start are brief discussion about vectors their definition, mathematical representation, operations, types, and their application in different fields of physics.
Definition, mathematical representation, operations, and their application in different fields of physics in detail are given below:
It can be defined as:
"The quantity which can described through both direction and magnitude is known as vector quantity or vectors."
In mechanics or geometry firstly term vector can be used but in some articles, the word or term vector can also be used for tuples. because mostly in mechanics which is the branches of physics vector quantities are used for magnitude and direction. some examples of vector quantities are given there:
Representation of vectors in detail is given there:
The vector quantity can be represented through the bold letter. For instance, the normal letter is v but for the vector, it can be written in bold form like v. Another example is the normal letter is written as a,b, and c but if it can be used for vector it can be written as a, b and c.
The vectors can also be represented by putting the arrowhead over the letter. Some examples are given there:
Vectors can also be represented in graphical form through an arrow. In the graphical representation, the arrow points to the direction of the vector, and the length of an arrow can represent the magnitude of the vectors.
For instance, a vector can cover the displacement from point A to point B then the arrow length represents the magnitude and the arrow point represents the direction from point A to point B.
The components of the vectors can also be expressed in the coordinate axis. Their components can be expressed in the two-dimensional cartesian coordinate system or the three-dimensional cartesian coordinate system. In a dimensional coordinate system, there are two axes x and y so the vector A in the two-dimensional system can be broken and written as AX on the x component and Ay in the y component.
But in the three-dimensional cartesian coordinate system x, y, and z are the three components and the vector A is written as Ax on the x component, Ayon the y component, and Az in the z component.
In the two-dimensional coordinate system, the vector A is mathematically written as;
A = AXi + Ayj
In the three-dimensional coordinate system, the vector A is mathematically written as;
A = Axi + Ayj + Azk
There i, j, and k are the unit vectors in the direction of vector components x, y, and z.
Various types of vectors are used in physics or mechanics some types of vectors with their details are given there:
Null or zero vector
Equal vector
Position vector
Negative of a vector
Like or unlike vector
Unit vector
Displacement vector
Coplanar vector
Co initial vector
Collinear vector
Their description is given there:
A null vector is also termed a zero vector. In vector is referred to as a zero or null vector when its magnitude is zero and there is no specific direction where the arrow points. In null or zero vector the length of magnitude is also zero. Their starting and ending points are the same. For instance, the vector OQ has the line segment and the starting point is O and the end is at the same point Q so their magnitude is 0.
Two different vectors are termed equal vectors if they have equal magnitude and also their direction. Rather they may have different starting points but their magnitude and the direction are same. For instance, if the vector magnitude is equal to the magnitude of vector b and their arrow is pointed in the same direction.
The vector can represent the origin and the position at any point related to the initial or origin point. Position vector can describe the direction of a vector from the main origin point to the endpoint.
The vector is termed as negative a vector if the vector which is given has the same magnitude and direction but at a point, any vector can change its direction means they have the same magnitude but have opposite directions so this vector which has the same magnitude but the opposite direction is known as negative of the vector. For example, vector A and vector B have the same magnitude but they have opposite directions and are written as A = -B
The vectors are termed as like vectors if they have the same direction but if the vectors do have not the same direction then they are called unlike vectors. For example, if the vector AB has the same direction then they are like vectors but if vector AB does have not the same direction then they are unlike vectors.
The vectors are termed unit vectors if they have only one magnitude. A unit vector is slo referred as the direction vector. The formulas that are used for unit vectors are:
V = VV
V represents a unit vector, V represents a vector, and V represents the magnitude of a vector.
If the quantity can be a displacement from point A to point B then the displacement between the AB is termed a displacement vector. For example, if a quantity is moved from point A and reaches point B then the distance between these points is termed a displacement vector it can also be termed the position vector.
The vectors that are placed in three-dimensional space with the same plane then it can be termed as the coplanar vectors. All vectors are parallel to each other in the same plane.
Two vectors are termed collinear if they are parallel to each other and not dependent upon the magnitude or direction. Collinear vectors are also termed parallel vectors. For example, if the vector A and vector B are opposite in direction and both have different magnitudes but they are parallel to each other then it can be called a collinear or parallel vector.
When two or many vectors have the same origin or initial point on the same plane then it can termed as the co-initial vectors. For example, the vectors A, B and C can originate from the same point with the same plane then it can be called as co initial vector.
Different mathematical operations with vectors are given below:
Addition of vector
Subtraction of vector
Dot product ( Scalar product)
Scalar multiplication
Cross product (Vector product)
For addition and subtraction of the vector, we can use the tip-to-tail rule in which the tail of the second vector can be placed on the tip of the first vector and the first vector tail is placed on the tip of the second vector.
For the addition of the vector we can use the tip-to-tail rule their mathematical representation is given below;
If we have the vector A their component on x or y is AX and Ay and the other vector B then their component on x and y is Bx and By and their sum is represented by C then they can be added as:
C = A+ B
Then,
A+ B = (Ax + Bx, Ay+ By)
Like the addition of a vector, the subtraction of a vector can also follow the head-to-tail rule. so the component method and mathematically expressed equation for the subtraction of a vector are given below:
If we have the vector A their component on x or y is AX and Ay and the other vector B then their component on x and y is Bx and By and their subtraction is represented by C then they can be subtracted as:
C = A– B
Then,
A– B = (Ax – Bx, Ay– By)
The cross product is also known as the vector product. In the vector product, the vector A and the vector B are perpendicular to each other then their product in mathematical form can be written as:
A B = ( AyBz– AzBy) i + (AzBx– AxBz) j + ( AxBy – AyBx) k
The magnitude of the vector or cross product is,
A B = ABsinθ
In this, the sinθ is the angle between the vector A and the vector B.
In physics cross products are used to understand the rotational and the circular motion of an object also they can be used to calculate precise calculations like torque.
When we multiply the vector with the scalar which has only magnitude and these quantities do have not a direction but if the scalar is negative then it means that their direction can be reversed. Their mathematical form is given there,
If we have the vector A and their component on x and y axis is Ax or Ay and the scalar is K then they can be written as:
KA = ( KAX , KAy)
The vector A and the vector B, their dot product is the scalar quantity so that's why they are also termed as the scalar product. Their mathematical expression is given below:
A B = AxBx + AyBy + AzBz
Their product can express their magnitude, not their direction. So the magnitude of the scalar or the dot product is,
A B = AB cosθ
cosθ is the angle between the vector A and the vector B for expressing the magnitude.
We can use the scalar and dot product in physics to calculate or determine the angle between the two vectors if one vector is projecting to another vector.
Vector quantities can play a very essential role in the physics for calculating numerous calculations. Vectors can be used in various fields of physics because by busying the vector mathematical operations we can do very precise and accurate calculations. Now we can discuss some applications of vector quantites in physics. The field of physics in which the vectors are used is given there:
In Mechanics
Newton’s law of motion
Electromagnetism
Maxwell’s equation
Bernoullis equation
Fluid dynamics
Quantum mechanics
Velocity field
Schrodinger equation
Their detail is given there:
To describe the motion, forces, and displacement in mechanics we can use the vector quantities. Some vector quantities explanation is given there:
Acceleration: acceleration is the time rate change of velocity, and the vector can express the change of direction and magnitude along with experiencing the quick velocity changing
Displacement: the position which can be changed by the moving object can be expressed by vectors. And the direction with magnitude of these objects can be expressed through vectors.
Velocity: velocity is the rate of change of displacement, their magnitude, direction, and speed can be represented by the vectors.
The equation of motion which can be used in the physics of the vector quantities can also involved in them. For example, the formula for the uniform acceleration is
v = v0 + at
There v expresses the velocity vector, v0 indicates the initial velocity, a is for acceleration and t expresses the time in which the velocity can be changed.
In Newton, the law of motion all formulas of Newton's law can be expressed by using vectors. The law of motion with its formula which indicates the magnitude and direction is given there:
The first law of motion: the object always remains at rest unless the external force can act upon it. And through these external forces, the object can be moved.
The second law of motion: the force that can act on the moving object is equal to the mass of an object and also equal to the acceleration. It can be written as F = ma where F is the vector quantity force, m indicates mass and a indicates the acceleration.
The third law of motion: is that every reaction has an equal and opposite reaction it can indicate the magnitude and direction of every action and reaction.
The dynamics of the object, its direction, and magnitude can be expressed through vectors in these equations of motion.
The vector spaces that can be involved in quantum mechanics are also termed as the Hilbert spaces. The state of particles is represented by the vectors which are used in quantum mechanics and it is also called wavefunction and state vectors. These vectors can also indicate the direction and magnitude of the particles in quantum mechanics.
In the field of fluid dynamics, the vectors represent the flow of fluid along with their direction and they can also represent the properties of the fluid which can flow. Vectors that can be used in fluid dynamics can also represent their magnitude with properties.
Schrodinger equation is the time-independent equation. This equation can be described and indicate the quantum state of any physical system independent of time. The vectors which can be used in these equations are written there:
iћ∂∂tψ (r, t) = H ψ ( r, t)
there, ћ expressed the Planck constant, ψ (r,t)expressed the wavefunctions, and the H expressed the Hamilton operator.
In electromagnetism, the vector quantities can expressed and describe the electrical field and the magnetic field. These vectors can also explain the relationship and the interactions of current and the charges in both electrical and magnetic fields.
The vectors that can be used to express the electric field are;
E = Fq
Where E is the vector that can indicate the electrical field in the space at any point, F is the force that can be experienced by the unit positive charges in the electric field and the q expresses the charges which are present in the electrical field.
The vector that expresses the magnetic field and the Lorentz force law is written there:
F = q (v B)
The magnetic field can be expressed through vector B and this vector B can describe the magnetic forces which are present in the magnetic field. The force F experienced by the charges can be expressed by the vector q with the same velocity which can indicated by v.
Bernoulli's equation can be derived and dependent on the vector. The vectors which are related in the Bernoullis equation are height, pressure, and velocity in the flowing fluid. The Bernoullis equation formula in which vectors are used is given there:
P + 12 ρv2 + ρgh = constant
There P represents the pressure, ρ represents the fluid density, h represents the height of the fluid according to the reference point, and v represents the fluid velocity.
The electric and magnetic field construction and how these fields can change their current and charges are described through the Maxwell equation. The vectors which can represent these fields current and charges are given there:
The vectors that can represent the current and charges in the field are given there:
⛛ E = -∂B∂t
The vectors that represent the Gauss law formula for electricity are given there:
⛛ E = pe0
The vectors that can represent the amperes law formula are given there:
⛛ B = μ0J + μ0ε0∂E∂t
The vectors that can represent the formula of Gauss law for magnetism are given there:
⛛ B = 0
Fluid particles that are present in space at different points can be explained and described through vectors like v (r,t), and in this v is the vector for velocity, r for the position of the particle, and t is for a time in which the velocity of the particles can be changed and dependent upon the time.
In the modern era of science, vectors can be used in many advanced topics in physics some topics with descriptions are given there:
Tensors
Gradient
curl
Vector calculus
Divergence
The complex physical quantities are also represented and described by the vectors and these vectors which describe complex quantities are generalized, known as tensors. In the continuum mechanics or in the theory of relativity tensors mostly tensors are used.
The vector field gradient is represented by ⛛ and the scalar field gradient is Φ. When the rate of Φ increased then the magnitude rate also increased and this gradient mathematically can be written as:
⛛Φ = (∂Φ∂x , ∂Φ∂y, ∂Φ∂z )
The ability and the tendency of the field that can be moved or rotated in a point is termed the curl of the vector field A. Mathematically it can written as:
⛛ A = (∂Az∂y – ∂Ay∂z , ∂Ax∂z – ∂Az∂x, ∂Ay∂x – ∂Ax∂y)
The concept of vectors in different fields of physics and mathematics can be extended through vector calculus because we can do different mathematical operations with vectors like curl, gradient, and divergence.
By using the divergence formula we can measure the rate of flow of vectors in the vector field of A. The mathematical expression and the formula of divergence are given below:
⛛ A = ( ∂Ax∂x + ∂Ay∂y + ∂Az∂z )
In physics, vectors are used as an essential tool because they can provide comprehensive information about the quantity and can also analyze and provide the description of the magnitude and the direction of the quantity in a very efficient way. Mostly it can be used in the field of physics like fluid dynamics, mechanics, electromagnetism, quantum mechanics, and in mathematical operations to derive or express the formula. Vectors can play a very essential role in physics or mathematics. Vectors can become the backbone of calculations in physics or in mathematics because they can help in doing very crucial calculations.
Hi friends, I hope you are all well. Today we can talk about the dimensions of all physical quantities which include the base quantities and the derived quantities. Dimensions of physical quantities are the fundamental part that helps us understand the physical and natural properties of any physical quantity. In modern science and technology, in engineering, and in different fields of physics where physical quantities units are used, the dimensions of these units help to derive a new formula and are also used in derivations. Dimensions of physical quantities also convey the detail of the types of physical quantity.
Dimension of physical quantities also helps to check the correctness of the equation and the formula that we can derive or use to solve the problems. Dimension of the physical quantities systematically expressed the physical quantities units. For analysis of the formula and the physical quantities, we can also use the dimensions of these quantities. Now we can start our detailed discussion about the dimensions of physical quantities which include the base quantities dimensions, derived quantities dimensions, and the application and analysis of the physical quantities in physics.
Physical quantities are defined as:
“The quantities which can be measured and quantified by measurements are termed as physical quantities.” For example mass, length weight, electric current, and many other various quantities. These physical quantities can be expressed in the algebraic form and when measured we can use different units for their measurements. To know the correctness of these units we can use dimension analysis.
For example for the quantity length we can use the unit meter and their symbol is m so by using the dimension of the meter we can verify them.
There are two main types of physical quantities units which are given below:
Base quantities
Derived quantities
The units that can be used to describe these physical quantities are known as base quantity units and the derived quantity units.
Dimension of physical quantities is defined as:
"The fundamental quantities which can be expressed in the form of raised power to describe the physical quantities are termed as dimensions of physical quantities."
The unit and the dimension of the physical quantity are written in the square brackets. For example, the unit of length is m and the symbol is m but their dimension is L.
The dimensions of base physical quantities and the derived physical quantities in detail are given below:
There are seven basic base physical quantities units which are also known as the building blocks units from which other units are derived. The dimensions of these base physical quantities with their units and definitions in detail are given there:
Base quantity |
Symbol |
Unit |
SI unit symbol |
Description |
Dimension |
Length |
L |
meter |
m |
The unit meter is used for the length and the length describes the distance between the two objects and also describes the height and the width of an object. |
L |
Mass |
m |
kilogram |
kg |
The quantity mass is used to measure the amount of matter of the objects. The measurements that can be measured are expressed in the unit kilogram. |
M |
Time |
t |
second |
s |
The physical quantity of time is used to measure the duration of the process and events like the duration of waves and oscillations. These measurements can be expressed with the unit second. |
T |
Electric current |
I |
Ampere |
A |
The flow of the electric charge in the electric circuits can be measured by using the term electric current and this measurement is expressed with the unit ampere. |
I |
Amount of substance |
N |
mole |
mol |
To measure the number of atoms, molecules, and other entities in the compound or matter we can use the term amount of substance the unit which is used to express the amount of substance is mole. |
N |
Luminous intensity |
J |
candela |
cd |
The power of light can be measured in terms of luminous intensity and these measurements are expressed with the unit candela. |
J |
Thermodynamic temperature |
Ө |
kelvin |
k |
The temperature in the form of heat which may be endothermic or exothermic released from the thermodynamic system is calculated or, measured and these measurements are expressed with the unit kelvin. |
Ө |
Derived units are derived from the seven basic base quantities units such as area, volume, power, and many others. Some of these derived units with their symbols and dimensions are given below:
Derived Quantities |
Symbols |
Formula and relation with other quantities |
Dimensions |
Dimensions related to the formula |
SI units |
SI units symbol |
Force |
F |
mass acceleration |
m= M acceleration = LT-2 |
MLT-2 |
Newton |
N |
Area |
A |
length breadth |
L= L breadth=L |
M0L2T0 |
Meter square |
m2 |
Density |
ρ |
mass volume |
m= M volume = L3 |
ML-3T0 |
Kilogram cubic per meter |
kgm-3 |
stress |
σ |
Forcearea |
F= MLT-2 A= L2 |
ML-1T-2 |
Newton per meter square |
Nm-2 |
Surface energy |
σs |
energyarea |
energy= ML2T-2 Area = L2 |
ML0T-2 |
Candela meter per second square |
Jm-2 |
Impulse |
J, imp |
force time |
force= MLT-2 Time= T |
MLT-1 |
Newton per second |
Ns-1 |
Strain |
ε |
change in dimensionorginal dimension |
Dimensionless |
—- |
—--- |
—--- |
Hubble constant |
H0 |
velocity of recession distance |
velocity= LT-1 distance= L |
M0L0T-1 |
Per second |
s-1 |
Coefficient of elasticity |
δ |
strssstrain |
stress= ML-1T-2 strain=1 |
ML-1T-2 |
Newton per meter square |
Nm-2 |
Volume |
V |
Length height breadth |
length= L height= L breadth= L |
M0L3T0 |
Cubic meter |
m3 |
Thrust |
N |
Force |
force = MLT-2 |
MLT-2 |
Newton per meter square or Pascal |
Nm-2or Pa |
Linear acceleration |
a |
velocitytime |
velocity= LT-1 time= T |
M0LT-2 |
Meter per second square |
ms-2 |
Work |
W |
Force distance |
force= MLT-2 distance= L |
ML2T-2 |
joule |
J |
Specific volume |
v |
volume/mass |
volume= L3 mass= M |
M-1L3T0 |
Cubic meter per kilogram |
m3kg-1 |
Specific gravity |
s.g |
density of material density of water |
—--- |
Dimenonless |
||
Tension |
T |
Force |
force= MLT-2 |
MLT-2 |
Newton meter per square and Pascal |
Nm-2 or Pa |
Surface tension |
Y |
force/ length |
Force= ML0T-2 length = L |
ML0T-2 |
Newton per meter |
Nm-1 |
Radius of gyration |
k |
distance |
distance= L |
L |
meter |
m |
Angular velocity |
ω |
angle/ time |
angle= L time= T |
LT-1 |
Radian per second |
rs-1 |
Momentum |
p |
Mass velocity |
mass= M velocity= LT-1 |
MLT-1 |
Kilogram meter per second |
Kg ms-1 |
Rate flow |
Q |
volume/ time |
Volume =L3 Time= T |
M0L3 T-1 |
Cubic meter per second |
m3s-1 |
Frequency |
λ |
No vibrations/ time |
Time= T |
M0L0T-1 |
meter |
m |
Heat |
Q |
energy |
energy= M1L2T-2 |
M1L2T-2 |
joule |
J |
Buoyant force |
force |
force= M1L1T-2 |
M1L1T-2 |
Newton |
N |
|
Plancks constant |
h |
energy/ frequency |
energy= M1L2 frequency= T |
M1 L2T-1 |
Joule second |
Js |
To derive the formula and the relationship between the different numerous physical quantities we can use a method or a technique known as dimensional analysis. Dimensional analysis can also be used to identify and determine the correctness of the equation and the formula and also change the units from one system to another systems. Dimensional analysis is based on the following points which are given there:
Principles of dimensional analysis
Steps involved in the dimensional analysis
Details of these points are given below:
Principles of dimensional analysis include:
Derivation of formulas
Dimensional homogeneity
Conversion of units
We can convert the units with the help of dimensional analysis in some other units. But the units that can we convert are expressed in terms of the base units. Some examples of conversion of the units are given there:
The unit square per meter which we we used to express the velocity is converted into some other base unit like km/h through dimension analysis.
If we can measure the distance in the units meters then we can convert it into the other base units like meters per hour.
The formula or the equation that we may derive and use for solving the problems has equal and the same similar dimensions on both sides which proves that the equation is correct. if the dimensions on both sides are not the same it proves that the equation is not correct. So dimension analysis also helps to identify or determine whether the equation or formula is correct or not. Some examples are given there:
For example, the formula or area is A = Length breadth. The dimension of the area is L2 and on the other side, the dimension of length is L and breadth dimension is L so on both sides dimensions are the same proves that the formula is correct.
Another example is a force equal to F= ma and the dimension of force is MLT-2 and on the other side the dimension of m is M and the dimension of a is LT-2. so it proves that the formula is correct because the dimensions on both sides are the same.
We can derive the different authentic formulas by using numerous physical quantities with the help of dimensional analysis. Dimensional analysis helps to understand the property of quantity which we can use to derive the formula. Some examples are given there:
We can derive the period of the pendulum by using the dimension of time and distance.
We can also derive the formula for the force by using different physical quantities like mass, acceleration, and time and we can also check whether the equation is correct or not with the help of dimensional analysis.
The steps which are involved in the dimensional analysis are given there:
Set up the equation
Identify the base quantity
Write the dimensional formula
Solve for the unknowns
For dimensional analysis, it is essential to check or confirm that the equation or formula we are using has the same dimensions on both sides and they are equal to each other. Some examples are given below:
The formula which is used for work is W= force distance the dimension for work is the same and equal to the dimension of force and the distance.
Another example is the formula of force is F= ma and the dimension of force is equal to the dimension of mass and acceleration so that's why the dimension analysis helps to identify the equation is correct.
In dimensional analysis, it is essential to identify the physical quantities that we are using to derive the formula or to solve the problems. Some examples are given below:
Identify the mass, length, volume, and other physical quantities and then use their dimension for dimensional analysis.
To identify the unknown physical quantities we can use the dimensions. And by the help of dimensions we can identify and determine the right physical quantity. Some examples are given below:
We can identify the gravitational constant G by using the dimensions and also determine their dimensions with the help of dimensional analysis.
For another example, we can determine the pendulum period by using the different dimensional equations, and after determining we can also identify their dimension with the help of dimensional analysis.
In the formula that we derived or used to solve the problems, it is essential to write the dimension of all quantities that can be used in the formula. Some examples are given below:
The dimensional formula of the area is L2.
The dimensional formula of power is Ml2T-3.
In the era of modern sciences and technology, engineers and other scientists measure complex quantities. Dimensional analysis helps to measure these complex quantities and also helps to derive new complex formulas and check the correctness of the derived formula by using the dimensions. Now in the modern era of physics and engineering, dimensional analysis can be used in many different fields. Some applications of dimensional analysis are given below:
Biophysics
Relativity
Plancks constant
Engineering
In biophysics, we can study and understand how the substance in the body or biological tissues can be moved or flow. Because diffusion mechanism can be used and we can measure and express them in some units so that's why we can use dimension analysis in it also
Some examples are given below:
The coefficient of diffusion can be described as the flow of material from one to another place and its dimensional formula is L2T-1.
To understand the relationship between mass and energy we can use the dimensional analysis method. Because they help to determine or identify the quantities that can be used in their dimensions. some examples are given there:
The formula which can be derived by Einstein is based on the relationship between the mass and the speed of light the formula is F=mc2 and the dimensions of m is M and the dimension of c2 is LT-1.
In quantum mechanics, we can understand the energy levels in the atoms and with the help of dimensional analysis, we can check the correctness of equations and detail understand the the energy levels in the atoms. Examples of dimensional analysis in quantum mechanics are given there:
For dimensional analysis and understanding the energy level we can use the formula of Plancks constant which is E=hv in this h is for energy level v is for frequency and their dimension is ML2T-1.
In engineering dimensional analysis is used for stress and strain because they can design the new materials and check their capacity to bear the load. So that's why dimensional analysis is used to determine the dimensions of stress and strain. Their example is given there:
In engineering, the stress on the object at force per unit area and strain changes the length and their dimensional analysis ML-1T-2.
Dimensional analysis plays a vital role in the modern era of science and modern technology because in this era we can measure complex physical quantities and checking the correctness of dimensional analysis is essential. It can also help to derive the new formula and understand the natural properties of the quantities. By using dimensional analysis scientists and engineers can achieve highly precise and accurate measurements of complex physical quantities.
With time when the physical quantities become more complex and developed then the dimensional analysis techniques can also be developed and adopt different changes in them. The steps that can included in the dimensional analysis can be developed and more steps are included for the analysis of complex physical quantities which helps the scientist to understand the quantities and give the most precise and accurate result according to complexity.
Hi friends, I hope you are all well. In this article, we can discuss the uncertainty in the measurements which can be measured. In the era of modern science and technology or modern physics, scientists can measure complex quantities and these measurements are not precise and accurate somehow doubt is present in these measurements, these doubts are suspicious known as uncertainty in the measurements. In physics or other fields of technology and engineering measurement is essential to measure or understand the quantity of a material or an object. Because every measurement is correct there are always some doubts or doubtful digits and they are called uncertainty in the measurements.
Now in this article, we can explore the history, definition, quantifying methods, and different techniques that can be used to minimize uncertainty and also explore their applications and significance in different fields of engineering and physics.
All substantial National measurement institutes can research the uncertainty in the measurements and give detailed documents about the measurement which is known as GUM and stands for the “Guide of Uncertainty in the Measurements.” this document gives the details about the uncertainties in the measurements. In metrology measurements when we can take measurements of the object many times it is confirmed that somehow measurements are not correct and precise. Or the doubtful measurements are termed uncertainty. All the measurements are not always correct because the measurement results depend upon the instrument's efficiency and the skilled person who can take measurements That is why uncertainty comes into measurements due to many various factors that may depend upon the environmental factors also.
Measurements are essential to determine the quantity of the physical quantity or any objects. Measurements also play a vital role at the economic level. The quality of laboratories can also determined by their calibration results because the exact measurements help to understand the quantity of the object. So that's why the ASME which stands for the American Society of Mechanical Engineers can present different standards for the uncertainty in the measurements. According to their standards, the measurements are done and engineers and scientists in laboratories can also measure the quality of different measurements.
Uncertainty in the measurements is defined as:
"The measurement which can we measured have lack of certainty and they have a great difference between the true value and the measurement value which be measured."
In simple words, the measurements we can measure have some doubts and their results do not according to the expectations and lack sureness are termed as uncertainty in the measurements. Uncertainty in the measurements is common because not every measurement is accurate or precise. For instance, we can measure the length which is about 6.7cm with a meter ruler but the true value range is about 6.62 or 6.75cm so the uncertainty in this measurement is approximately 0.05cm. Another example is if we measure the height of an object and the measurement which we can measure is 5.5m but the true value range is 5.3m or 5.7m so in this measurement the uncertainty is approximately equal to 0.02m.
There are two major types of uncertainty in measurements which are given below:
Type A uncertainty
Type B uncertainty
Type A uncertainty is defined as:
"The uncertainty measurement which can be evaluated through the different methods of statical analysis are known as type A uncertainty measurements."
Generally, in type A uncertainty measurements we can measure or collect different data about the measurement and then observe the series of collected data, and then evaluate the uncertainty which are present in these measurements.
For example, we can take measurements of an object many times or maybe 20 times and then evaluate and observe the results of these measurements and then analyze the uncertainty in these measurements. from empirical data, we can directly identify or determine the type A uncertainty in the measurements. Another example is when we want precise and accurate measurements then we can measure the same measurements many times like if we can measure the length and the measurements range between 5.7cm to 14cm and in between different measurements occur after these measurements we can observe or estimate the average uncertainty.
Type A uncertainty can be measured by repeated measurements of the measuring object and evaluated through statistical methods or techniques. Some statistical methods that are included in the evaluation of type A uncertainty are given below:
Confidence intervals
Arithmetic means
Standard error of the mean
Degree of freedom
Standard deviation
Confidence interval is defined as:
"The true and standard values that can be measured through the measurement of quantity, and confidence interval convey the range of accuracy with a confidence level of the true measured value."
This formula can be used for a normal distribution which has a 95% confidence interval. The formula is given there:
μ z SEM
μ z SEM
There,
z is for the confidence level which can we desire and it is approximately equal to 1.96 for the normal distribution with a confidence level of 95%
Arithmetic means is defined as:
“The set of measured numbers, the all measured numbers are added and dived by the total numbers which are present in the set and the central number which is present in the set also added with the all measured numbers”.
To know an average of all measured measurements we can use the arithmetic formula because, through this statistical technique, we can evaluate the type A uncertainty in the measured sets.
The formula which is used to calculate the average of the measured set is given there:
x = 1n i=1nxi
The standard error of the mean is defined as:
“ the uncertainty which is present in the average that can be calculated from the set of measurements with no of measurements and the standard deviation, these estimate can be conveyed by the standard error of the mean.”
The formulas that are used for calculating the standard error of the mean are given there:
SEM= σN
there,
SEM = standard error of the mean
σ= standard deviation
N = no of measurements
Standard deviation can be defined as:
“The average can be measured from the dispersion of the set of collected measurements.”
Standard deviation can be used to measure the average variance which can be essential for evaluating the Type A uncertainty.
The formula that can be used for the standard deviation or measure the average variances is given there:
σ= 1n-1i=1n(xi-x)2
The degree of freedom can be defined as:
After the calculation of the standard deviation and the average the final numbers can be calculated freely for statistical analysis and the final values can help to understand the type A uncertainty and the degree of freedom.
The formula that is used for the degree of freedom is given there:
v= n - 1
The main causes and sources of the Type A uncertainty in the measurements are given there:
Environmental changes
Human factors
Instrumental fluctuations
The methods that can reduce the uncertainty in the measurements are given below:
Advanced measurement techniques
Repeated numbers of measurements
Control environmental factors
Improving measurements techniques
Type b uncertainty can be defined as:
"the uncertainty which can be evaluated by using different methods except the statistical analysis of measurements. Type B uncertainty can't be evaluated through statistical analysis they can be evaluated through calibration certificates, scientists' judgment, and through the publishers."
Type B uncertainty can be measured differently from Type A uncertainty in the measurements because it is mostly evaluated through the collected information and through the publishers. This type of error is also common the main sources and the causes of these uncertainties can be explained below.
The methods that can be used to evaluate the type B uncertainties in the measurements are given below:
Expert judgment
Manufacturer specification
Theoretical analysis
Reviews of calibration certificates
Reference material data
Chats of collected data information
When we can do measurements but do have not direct measurements and data then the uncertainty measurement is provided to the experts who have experienced and understand the limitations and uncertainties of the measurements. Then the judges understand the uncertainty and then identify the type of uncertainty and try to reduce these uncertainties. Some examples are given below:
The scientists who can do experiments and want to change into theory the council and the judges understand the experiment according to their experience and then allow them.
A well-experienced scientist or metrologist can measure the uncertainty in the measurements through their experience and knowledge.
The type B uncertainty can be evaluated by using the calibration certificate because the calibration certificates convey information and details about the accuracy and precision of the measuring instruments. Calibration certificates also provide information about the correction of uncertainty in the measurements.
The uncertainty of the voltmeter is 0.05 provided in the calibration certificate and how to recover this uncertainty information is also present in it.
The uncertain information and documents are provided through the reference material data. The information and values that are provided through these reference data help to improve the uncertainty in the measurements that can be calculated.
The certified uncertainty of the gas analyzer is approximately about 0.1% and the reference data is provided to minimize the uncertainty in the measurements.
The instrument's specification accuracy, precision, and limits can only defined by the manufacturers because they understand the nature of their instruments and they also determine or estimate the uncertainty that can produced by their instrument during the measurements.
For the dimensional measurements the instruments we can use a digital micrometer and their accuracy is about 0.002 it can also budget the uncertainty measurement in it.
There are different theoretical models are present that can convey detailed information about the uncertainties in the measurements. Because these models are based on the assumptions and the practical experiences. By using these models we can also estimate and identify the uncertainties in the measurements which can be measured.
We can estimate the uncertainty and the precision in the vacuum of the speed of light and we can measure these uncertainties that are based on precision and we can calculate them.
The sources which can use the uncertainty in the measurements are given below:
Environmental conditions
Previous measurements
Manufactures specification
Theoretical models
Instrumental calibration
Previous measurements
In the combination of different type b components, we can use the root sum square method to estimate and calculate the uncertainty in the measurements. The all components that are combined are independent but we can combine them to estimate the uncertainty in the measurements precisely.
uc(y) = i=1nciu(xi)2
We can reduce the type B uncertainty in the measurements if we can follow these given steps. Because it can help to reduce tp understanding of the uncertainties in the measurements. The methods and the steps are given below:
Used advanced measurement techniques
Improved calibration
Used high-quality reference data
Enhanced environmental controls.
Generally, uncertainty in the measurements occurs due to many sources but the major two are
Random error
Systematic error
The sources and details of these errors are given below:
Random errors are common because they can caused by many different sources and they may be reduced by doing repeated measurements and by estimating the main cause of error. Some major sources which can cause this error are given there:
Observer variability
Environmental noise
Instrumental fluctuations
This error occurs due to the imperfect instruments usage and the unskilled persons who can take measurements but the main sources and causes of systematic error are given there:
Methodological error
Instrumental error
Observer error
Environmental factors
The techniques and some methods that are used to reduce the uncertainty are given there:
Replicates and repeat measurements
Randomization
Calibration and standardization
Improved experimental designs
Control variables
To reduce the uncertainty in the measurements we can use many different advanced techniques some are given there:
Error-correcting algorithm
Automated data collection
High precision instruments
Skilled persons
In scientific research the measurement of uncertainty is essential and it is also essential to reduce it because the scientists try to make precise and accurate measurements according to the calibration certificates, the significance of uncertainty in scientific research is given there:
Reproducibility
Peer review
Validation
Transparent reporting
In the field of modern technology and engineering, in measurements uncertainty and error are common but by using different techniques we can reduce them. Some applications of uncertainty in the management are given there:
Medical and biological research
Pollution monitoring
Quality control
Climate modeling
Safety standards
Drug efficacy
Diagnostic accuracy
The uncertainty in the measurements is common but in modern science and technology or different fields of science and physics, we can reduce the uncertainty in the measurements using many different techniques because scientists and engineers want to measure the precise and accurate measured values. Because the experts can agree on the measurements which are according to the standard values of the calibration certificates. Because the national measuring institutes and the American Society of Measurements can present the standard suits for measurements that are used to reduce or estimate the uncertainty in the measurements.