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.
Hello friends, I hope you are all doing well. In this article, we can talk about the precision and accuracy of the measurement. Both of these are used to analyze errors in the measurements which can be calculated. In the era of science and modern technology, accuracy and precision are essential to depict the quality of the measurements. To measure the error in the measurement precision and accuracy techniques can be used because both are used for comprehensive measurement analysis.
To calculate or describe the random errors precision of measurement can be used but if we want to describe the systematic error, accuracy of measurements can be used. Now the details of precision and accuracy, their definition, differences, examples, and their application in different fields are given below:
The main differences between precision and accuracy are given below:
| Precision | Accuracy |
Definition: The reproducibility which can be measured in the measurement is known as precision. On the other hand, precision can also be defined as the set of repeatability measurements and all measurements are close to each other but we precise the closest value. Because highly precise measurements can convey the best and very similar results. |
Definition: The value that we can measure is close to the true value of the measurement is known as accuracy. Another definition of accuracy is that the value that can be measured is approximately equal or close to the accepted and the true value of the measured object. Because accuracy provides the correct measurement of the object which can we measured. If the measurement that we can calculate is close to the standard value of measurement then it can be an accurate measured value. |
Accurate: The measurements that can be calculated are precise but they can't be accurate because high precision can cause low accuracy. |
Precision: The measurements that we can calculate are accurate but they are not precise because high accuracy affects the precision and causes low precision measurement.
|
Determined: To determine the precision we can calculate the object measurement many times and from a set of measured measurements we can determine the precise measurement value. |
Determined: Accuracy can be determined from a single measurement of an object which can be measured. |
Random error: Random error can affect the precise measurement which we can determine. |
Systematic error: The accuracy of the measurement can be affected by systematic error. |
Accurate: The measurements that we can measure are precise but not correct every time. |
Precision: The measurements that can be measured are accurate but these measurements are not correct anyway. |
Reproducibility: Precision can give a degree of reproducibility. |
Conformity: Accuracy provides the degree of conformity of measurements. |
The terms precision and accuracy are different from each other, but somehow they are related to each other. Because when we measure the object or any quantity measurement then it may be accurate but not to be precise but if we determine the periscred measurement then it not to be accurate so that why both terms are not present in the measurements at the same time.to understand the relation and the difference between the accuracy and measurement and to improve the measurement techniques some common points are given there:
When the measurements that we can calculate are close to the standard or true value but the measured values are not similar to each other then they have high accuracy but low precision because the measurements are not matched to each other. Some examples are given below:
The measurement of the field which we can calculate is 80 cm, 99.90 cm, and 100.02cm but the true or standard value is 100.10 cm so the values are accurate but they can’t be precise.
If we can calculate the length of an object and the measured value is 70.01m, 80.67m, and 90.01m, the true value is 80.67 then the all values are not precise but they are accurate.
The measured values that can be calculated through measurements are not close to the true value and also the set of measurements are not close to each other all the measurement values are different which means they have low precision and low accuracy. Some examples are given below:
The measurements of the yields are 75.00 cm, 100.00 cm, and 112.00cm but the true value is 120.00cm. These values are not close to each other and neither one value is close to the true value so that is why they have low accuracy and low precision.
For instance, if the measurement is 95.00 cm, 106.00 cm, and 101.00cm but the true value is 105.07cm then they have both low accuracy and low precision.
If the measured values that we can measure are close to the true and the standard value and the set of measurements that we can calculate are close to each other then we can say that the measurements which we can be calculated have high accuracy and also have high precision. Some examples of highly accurate and highly precise measurements are given below:
The measurement of the length of the cylinder is 101m, 102m, and 103m and the true or standard value or accuracy is 102m then these values are precise and close to each other and they are also accurate because the measurement is close to the true value.
The measurement of the yield is 80cm, 82cm, 83cm, and 85cm or the true value is 83cm then they have high accuracy and high precision because the values are close to the true value and all sets of measurements are close to each other.
The measured measurements are similar or close to each other but these measurements are not close to the true or standard value. some examples of these measurements which have high precision but low accuracy are given below:
The quantifying of accuracy and precision in detail are given below:
By using the numerous different statistical measures we can quantify the accuracy. The steps which are involved in quantifying the accuracy are given there:
Root mean squared error
Absolute error
Mean absolute error
Relative error
Their details are given there:
When we take the square roots of the average squared errors then it is known as root mean squared error. The formulas which are used for root mean squared error are given below:
RMSE= 1ni=1n(measured valuei-true values)2
Absolute errors are used to determine the difference between the true value and the measured value. The formulas which are used to determine the absolute error are given below:
Absolute error= measured value- true value
In the set of measurements to calculate the average error in all the measurements mean absolute error formulas are used. The mean absolute error formula is given there:
MAE= 1ni=1nmeasured value- true value
To express the true value percentage with the absolute error we can use the relative error formula. The relative error formula is given there:
Relative error= (absolute errortrue value) 100
By using the measure of variability we can quantify the precision. the steps which are included in the measure of variability are given there:
Coefficient of variation (CV)
Range
Variance
Standard deviation
Details of these points are given below:
Range is used to measure the difference between the maximum and the minimum values in the set of measured measurement values. The formula which can be used to measure the range is given below:
Range= Maximum value- minimum value
By using the mean values from the set of measurements to can measure the dispersion of the measurement set. The formulas which are used in standard deviation are given below:
Standard deviation: 1n-1i=1n(measured value-mean value)2
The square root of standard deviation is termed variance. The formula of variance is given there:
Variance= 1n-1i=1n (mmeasured value- mean value)2
To express the ratio of the standard deviation we can use the formula of coefficient of variation. The formula for the coefficient of variation is given there:
Coefficient of variation: standard deviationmean value 100
The classification of the accuracy and the precision are given below:
To measure the statistical measurements we can use accuracy and we can use accuracy in the binary classification to determine or identify the conditions of measurements. By using the accuracy to can determine the exact ratio of the true positive or the true negative.
The two main types of errors that affect the precision and accuracy but these two types of error have also types two main types of errors are given below:
Systematic error
Random error
Details of these errors with their types are given there:
Systematic error is defined as an error that occurs many times due to faulty equipment or may be due to an unskilled person's measurement techniques and many other reasons. But this type of error can directly affect the measurement which can be measured and give the inefficient or bad results of the measurements.
The sources that can cause the systematic error are given below:
Observational errors
Environmental errors
Theoretical errors
Instrumental errors
Observational errors occur due to human mistakes. When they can read the reading and observe the reading efficiently then observational errors occur some examples are given below:
The reading that can be measured by a human is approximately 3.4cm but it can't observe the certainty and uncertainty within it and also can’t read the exact digits of the reading.
When the unskilled person tries to observe the readings of very small quantities then the observational error occurs maximum because to measure the small quantities skilled person is used to observe the readings.
The error occurs due to environmental factors like temperature, humidity, and the fluctuation in the electromagnetic interference and also due to the airflow which can move instruments during the measurements. Some examples of environmental errors are given below:
When we use the thermometer to measure the temperature changing continuously in temperature can cause errors in the measurements.
To measure the current when we used the galvanometer the needle of the galvanometer moved again and again due to the magnetic field these fluctuations can cause errors in the measurements.
The error occurs due to the usage of imperfect or faulty instruments during measurements.
To measure the measurements without error firstly perfect instruments are essential because without this the measurement can't be correct when we measure anything. Some examples of instrumental errors are given below:
To measure the length of an object a ruler can be used but the readings on the ruler are visible clearly because it can be helped to measure the exact length.
A weight machine can be shown the weight of 2kg even if we can't put any object on it this is the instrumental error.
Theoretical errors occur when we assume the theory model but nothing can happen in reality according to these models. Theoretical misassumptions can cause different errors in the measurement and provide the un accurate or precise results. Some examples of theoretical errors are given there:
During physics quantities measurements the air resistance is considered negligible.
These errors occur due to various small unpredictable variables and these errors can’t be controlled because they occur in all measurements due to many different small or uncontrolled able variables. Some sources which can cause the random errors are given below:
Environmental variability
Instrumental vartibility
observer variability
Sample variability
The fluctuations that can be occurred in the environment at any time like fluctuations in the temperature. Some examples are given there:
The flow of air can distract the measuring instrument from its place and cause errors in the reading
Air currents can also cause errors in the measurements.
The changing that occurs in the instrument during the reading there are many different instruments present that can use the noise or changes in it during measurements. Some examples are given below:
During measuring the current the galvanometer changing the needle again and again can cause errors in the readings or measurements.
When the numerous observer measured the measurement at the same time their readings were different from each other due to sight issues and many other issues. Some examples are given below:
When the numerous observed reading through the stopwatch and stopwatch couldn't stop at the same time slime gave them vision issues.
When we can measure the same object but the different samples have changes naturally we can measure them. These natural changes can also cause errors in the measurements. Some examples are given below:
When taking different samples of the same thing or object naturally their width or thickness can be changed then different measurements come with errors.
The accuracy and the precision can be improved if we can follow the following steps which are given there:
For improving accuracy:
Environmental control
Calibration
Instrument maintenance
standardization
For improving precision:
Training
Environmental control
Reptation
Instrument quality
Application of precision and accuracy in different fields are given below:
Scientific research
In modern technology and sciences
In measurements
Healthcare fields
Monitoring environment
In mathematics calculations.
Precision and accuracy both play vital roles in measurements because in modern physics scientists can measure very small quantities and these quantities of measurement can be accurate without error due to the accuracy and precision. Both terms are the fundamental p[art of measurement in the analysis of comprehensive measurement.
Precise and accurate measurements provide the quality of the measurements because without these terms in measurements, too many errors occur and they are not precise and accurate.
Hi everyone, I hope you are doing well. Today in this post we can discuss significant figures. Significant figures can be evolved and traced a few hundred years ago and they can be developed from time to time according to the precise calculations that can be done in mathematics and modern science and technology.
The origin of significant figures can be traced to ancient times in Greece, Egypt, or many other countries where they can be used as Significant figures in calculations or mathematics. With time these significant figures can be used by astronomers, mathematicians, and scientists as well. To measure accurate and precise measurements they can use significant figures with scientific notation. In modern physics and mathematics, significant figures can be used essentially.
Now we can start our detailed discussion about significant figures and their definition, rules, examples, and problems in detail.
Details of significant figures are given below:
Significant figures can be defined as:
"In the number or calculated value the specific digits which provide precision and accuracy and also impart and convey the necessity of digits in the calculated measurement."
Significant figures include the zero or the nonzero digits. Significant figures are also known as significant digits and sig figs. When we make measurements or calculate the measurement such as when we measure length, pressure, and volume and the measured value is large then we can use significant or scientific notation method. or the most capable digits are estimated and called significant figures.
Such as if we measure the volume of a cylinder and the measurement is approximately equal to 3.97L and the uncertainty is approximately equals to 0.04L. The actual volume of this cylinder is approximately between 3.93L and 1.01L, but we can't know the certain digits in this measurement completely. But all of these three figures are called significant figures.
Additionally, the value 345.0 has only 3 significant figures 3,4, and 5 the o digit is not termed as a significant figure because it is only a placeholder.
Some digits are not to be considered as a significant figure. So the types of digits that can't be considered significant figures are given below:
Spurious
Leading zero
Trailing zero
The zero can only be present in the value or measurement as a placeholder. For example 1800 the last two zeros were only a placeholder but we can write the scientific notation we can raise the power of ten but in significant value, we only estimate and choose the figures that can provide accurate and precise measurement.
The zero which can be present at the start of measurement and the value is known as leading zero. For instance, 00367 the first two zeros (00) known as leading zero and it can’t be a significant figure in this value 3,6, and 7 are the significant figures. Another example is 0.00456089 in this value the leading zero is not a significant figure but the zero which is present in between the digits like 6,0,8 and 9 is referred to as a significant figure.
Spurious digits are those when we do calculations on the instruments then the high-resolution measurement results precisions and the accuracy digits are known as spurious digits.
Another example is if the zero is present in the value after the decimal it can be considered a significant figure like 12.00 then these two zeros are also significant figures and in this value, the total no of significant figures is 4 but if the decimal is not present like 12300 then the last two zero is not considered as significant figure and in this value, the total significant digits are 5. After the decimal point, the zero is also considered as the significant figure for example 1.000 the last trailing zeros are also the significant figures and the total significant figures in this value is 4. But if the decimal is not present like 1000 then this value has only one significant figure and trailing zeros are only a placeholder.
Some examples of significant figures are given below:
123790: All digits in this value are known as significant figures.
0.0056:in these digits the leading zero is not a significant figure the last two 5,6 are significant figures.
34986097: all digits are significant figures and the zero which is present inside them like 6,0,9,7 all are significant figures.
1001: all digits are the significant figures.
To identify the significant figures from the measurement and for the value, there are some rules. The rules that can help to identify the significant figures are given below:
Non-zero digits in the measurement are to be significant
Between the two significant digits, the zero is also a significant figure.
Starting, leading zeros are not a significant figure.
Zero after the decimal point on the right side can also be a significant figure.
Trailing zero without the decimal point is not significant.
All non-zero digits in the measurement or the value are significant figures some examples are given below:
9144 all the digits in this value are significant. So the total significant digits is 4.
1.5678 all the digits after decimal or before decimal are considered as significant figures and the total significant digits in it is 5.
568.9 these all digits are significant figures.
The digit zero is not to be considered to be significant but in between the two the significant digit zero also becomes a significant digit some examples are given below:
10054 in this value all digits including zero are referred to as significant figures.
102.60809 these all digits are considered significant figures the zeros after the decimal point are also considered significant figures.
101.101 in this value all digits are also considered as the significant digits or figures.
The starting or the leading zero is not considered to be a significant figure it can only considered as a placeholder some examples are given below:
000102 in this value last 3 digits 1,0and 2 are called significant but the starting three zeros are not considered to be significant but it is only placeholder or leading zeros.
0.065 the last two digits 6 and 5 are sig figs but the starting zero or zero after the decimal point is not considered a significant figure.
The trailing zero, the zero on the right side after the decimal point can considered a significant figure. Some examples are given below:
1.89900 these all digits are considered to be significant digits. In this value, the digits 1,8,9,9,0,0 all are significant figures including the trailing zeros.
0.1200 in this value 4 digits are called significant figure 1,2,0 and 0 are significant figures which are trailing zeros but the first leading zero is not considered as the significant figure.
The trailing zeros without the decimal point are not considered as a significant figure. some examples are given below:
19900 the first 3 digits 1,9 and 9 are considered as the significant digits but the last trailing zeros are not to be considered as the significant figure.
1096800 in this value the first 5 digits are considered to be significant but the last trailing zeros are not a significant digit or a figure.
The rounding-off technique can be used in mathematics in all calculations. In this technique the last digit if it is greater than 5 then it can be added to the previous number. for representing the rounding off number we can use n. To round the significant digit with n there are some rules which are given below:
In the measurement value if the last digit is greater or equal to the 5 then we can +1 inti the previous digit. For instance, 1.345 then after rounding the digit the new value is 1.35 and it has 3 significant figures now.
Another example is if we have 5 significant figures like 5.9867 then after rounding off the last digit the new value is 5.987 and they have only four digits in it.
Now if we have the last digit less than 5 then it can’t be added to the previous number some examples are given below:
1.563 the last digit is less than 5 so it can’t added 1 to the previous digit and the significant figure in this value is 4. Another example is that 67.91 has the 4 significant figures and they can’t be changed because the last digit value can’t be added.
To calculate the numerical value of significant figures for nonzero significant figure x and for the precision and accuracy p we can use the formula which is given below:
10n. Round (x10n)
Where the n is equal to,
n=[log 10(x)]+1-p
In the measurements precision and accuracy, both are essential. Generally, we can use precision for the stability of the measurements with repeated values and the accuracy used for the closeness to the actual and accurate measurement of the thing and the object. but with time in modern technology and science, both precision and accuracy are termed for trueness or the most closeness exact value of the measurements. In the scientific community, the accuracy and precise measurement of the object is essential because without accuracy the measurements and problems become more complicated That is why accuracy and precision both are essential for measurements.
In addition and subtraction, the result of measurements can calculated with the same decimal point before and after the addition and subtraction for example:
12.56+ 0.5=13.1 they have the 3 significant figures in it another example is that
67.9+9 =76.9 and the significant figure in it is 3.
The result that can be taken from measurement and the result after division and multiplication of values both have the same significant figures some examples are given below:
4.56 1.4 = 6.384 and we should rounded to 6.4 then they have 2 significant figures like before the division. another example is that 2.44 then it is equal to 9.6 they have also 2 significant digits like before the multiplication or division.
In the measurement, certainty means that we confidently know the significant digit in the measurements. This known digit conveys and provides the precision of measurement and is also considered reliable because accuracy and precision are very essential in the measurements.
For example, if we can measure the length of an object with a ruler that has a millimeter marking and the measurement is equal to 24.87 then the digits 2,4,8, and 7 are certain because measurement tools can provide precise and accurate measurements.
Rather uncertainty in the significant figures means that which measurements we can take are not precise and accurate and we have doubts about them. When an unskilled person or limited measurement tool is used for measurement then chances of uncertainty increase because a skilled person can take accurate and precise measurements of the object and the high precision measurement tool is also needed.
For example, if we can measure the length and the measurement is 24.56 then the last digit 6 creates uncertainty because it can be rounded off and the original length can't be measured precisely and efficiently. So that's if we want to remove uncertainty then we again measure the length and then determine the significant figures in it.
Some examples of significant figures are given below:
406.62 all digits in this are significant figures and the total significant figures is 5.
0.00034 the last two digits are signification figures and the starting zeros are leading zeros.
12090 all are significant figures and the last zero is only a placeholder.
74.0 the zero after decimal is also a significant figure the total significant figure is 3.
67.08 in this all digits are significant figures and the total is 4.
These are some common examples of significant figures.
Some practical problems to understand the significant figures are given below:
Identify the significant figures in the given problems:
7696
0.0074
690.00
60.09
74.0
Solutions to these problems are given below:
All 4 digits are significant figures.
Leading zeros are not significant digits but the last 2 digits 7 and 4 are significant.
All 5 digits including zero after decimal are also considered significant digits.
6,0,0 and 9 all digits are considered significant figures.
7 and 4 including trailing zero after the decimal are also considered significant figures.
The calculation to determine or calculate the significant figures some problems are given below:
26.7+ 8.1
62.4 0.04
2.4+ 9.2
58.0+ 4.5
0.056/0.0007
The solution to these problems is given below:
34.8 in this value 3, 4, and 8 are significant figures.
2.496 in this value 4 significant digits are present.
11.6 in this solution all digits are considered significant figures.
62.5 in this solution of a problem all digits are significant digits.
80 are the solution and both are significant figures.
To determine the significant figures, perform the following operations:
(2.7+ 4) 7.9
(5.6 8.9) + 8
Solutions to these problems are given below:
(6.7) x 7.9
=52.93 In this solution, all 4 digits are significant figures.
(49.84) + 8
= 57.84 In this solution, all 4 digits are significant figures.
Hi friends, I hope you all are well. In this post, we will talk about scientific notation. Scientific notation is an essential tool that scientists and engineers can use. Archemdies presented the idea of scientific notation in the 3rd century BC. His work and scientific notation ideas are based on the novel of time, known as place value.
Scientific notation ideas have evolved over many centuries but are finally represented by Archimedes. In the 16th or 17th century, mathematicians continued to adopt changes in them and invent many other new notable contributions like Rene Descartes who developed algebraic notation. Scientific notation is a way in which we can express large number values in short form in the form of an exponent or a decimal form.
Scientific notation not only simplifies the equation, but it can also help scientists and engineers to do calculations easily. In the United Kingdom scientific notation is also known as standard form or standard index form. Scientific notation is commonly used by scientists and on scientific calculators it is referred to as “SCI” display mode.
Scientific notation in detail is given below:
Scientific notation is defined as:
"it is the simplest way to express the large number into the small number or into in decimal form."
It can used by scientists, mathematicians, and or engineers to simplify the calculation and express the calculation in a very efficient way. In scientific notation, the number is expressed in the product form and they have two factors.
Generally, the scientific notation is written in:
N=m 10n
There:
Some examples of scientific notation are given below:
Covert 89,700 to scientific notation.
Firstly move the decimal to get 8.79
The decimal moved 3 places to the left, then the exponent is 8.
Then the scientific notation is 8.79103
Convert 0.0000023 into scientific notation:
Firstly move the decimal to get 2.3.
The decimal moved 6 places, then the exponent is 2.
Then the scientific notation is 2.3106.
We can write any real number equation in the form of m10n in various ways. In the normalized scientific notation, the value of m is 1 or greater than 1 but it is always less than 10 and the value of n depends upon the larger value or decimal place. thus equation 550 can be written as 5.50102. this scientific notation helps to compare the number easily in an efficient way if the exponent value is large that means that the number is normalized as compared to smaller exponents. The order of magnitude can be assumed when we subtract the exponent by separating the numbers.in the table of logarithms, we can use this form of numbers to solve the log questions. The exponent in the normalized equation is negative if the value ranges between 0 and 1 for example: 00.6 then it can be written in 610-2. the exponents may be equal to 10 if the real number value is large like 0.0000000009 then it can be written as 910-10.
In many fields, typically or generally normalized equation is used to express large numbers of values into simple and efficient decimal form. Exponential notation is also referred to as normalized notation. But in different fields, unnormalized or simple notation or equations can be used. Generally, the latter term of notation is more used where the value of m is not suppressed in the range between 1 to 10 and the base of the exponent may be changed or other than the 10.
Generally, calculators or computer programs use small or large number values to present scientific notation or in some calculators, all the numbers are present or configured uniformly.
The capital letter “E” or the small letter “e: can be used to represent or express the exponent which means “ten raised to the power of”. Typically in all computers or programs scientific notation or normalized notation can be abberived or represented in different styles. But the equation men is the scientific notation like m is the coefficient and the e is base 10 or n is for the exponent and it can be written as men or m10n. For example, 1.000 can be written as 1103 or also written as 1e3.
On the scientific calculator, the engineering notation can be written or expressed as “ENG”. The engineering notation is different from normalized notation because the value of exponent n is suppressed in normalized notation. Thus the value of m ranges in 1 ≤ |m| < 1000, other than 1 ≤ |m| < 10. So that's why sometimes the engineering notation is also known as scientific notation.
In engineering notation, the exponent value can be orally read by using prefixes such as nano, pico, Giga, and others. For example, 14.510-9 can as read as fourteen-point-five nanometers and can also be written as 14.5nm.
The number of digits in the value is used to find the degree of accuracy and precision in a value.
Significant figures include all nonzeros or zero digits but the first and the last zero digits are not significant figures if zero is present in between the numbers like 1,2,3,4,5 and so on then it is also called a significant figure. Some examples are given below:
456709800 In this value 7 significant figures are present and at the last two zero is only a placeholder and not referred to as significant figures.
Then 008809 in these values there are 4 significant figures present and the starting first two zeros are not called significant figures.
So when we convert 123459900 values into normalized or scientific notation then all significant figures are present and the nonsignificant figure is removed as 123459900 has 7 significant figures then it can be written in scientific notation as 1.234599108 or also written as 1.23459900108. thus the advantage of scientific notation is to signify the significant figures efficiently.
In scientific measurement, it is essential to record or measure all known digits from the measurement which can be calculated, and then estimate one or more digits if any information about this digit is acquirable. The estimated digits provided more information and became a significant figure because the estimated digits can provide more accurate and precise information about the leading and the ending digits.
The information that we can take from the estimated digit can help in the notation. This information also helps to choose the final digits or digits that are present in the value. For example, the estimated mass of a proton can be equal to 1.67262192369(51)10-27 kg. But in this value, the error occurs when we estimate the digit inefficiently and can be written as 5.110-37 and this estimated digit can increase the standard error or some other possible errors.
The rules that are used to convert the number into scientific notation are given below:
Identify the coefficient
Determine the exponent
Base
Non-zero integer
Positive or negative integer
Add or subtract the coefficient
Arithmetic operation with scientific notation
When we move the decimal point in the value then the original number which becomes the coefficient a is taken like 230000 then the decimal moves in it and becomes 2.3 and the 2 becomes the coefficient in the scientific notation.
Count the number of decimals that we moved in the value and this count becomes the exponent. For example: We have a value of 46000
Then it becomes, 4.6
The total no of decimals that we moved in the value is 4 then the exponent is 4 the value becomes 4.6x104.
The base of the exponent in the scientific notation is always 10.
The exponent is always a value it is always a non-zero integer like 104 or 108.
When we move the decimal from right to left then the exponent integer is positive. Example:
46000 then 4.610+4
When we move from left to right then the exponent integer is negative. Example
0.00677 then 6.7710-3
If the exponent is the same then we simply add or subtract them. Because the same exponent can be added or subtracted and also we can multiply or divide with them. For example:
3.4106 + 9.7106
The exponent is the same then we add them and then rewrite
3.4+9.7=10.04
Then rewrite them into scientific notation
10.04106 then the new scientific notation is
1.004107
In the division method, we divide the coefficient and subtract the exponent. An example is given below:
81062104= 8106-42
=4102
To add or subtract the scientific notation, the same exponent must be necessary. And adjust the equation if needed. An example is given below:
= 2.6103+ 4.0104
Then firstly we adjust the equation to same the exponent of both equations,
2.6103= 0.26104
Then add or subtract them,
=0.26104+ 4.0104
=4.26104
When we multiply the scientific notation then we add the exponent and multiply the coefficient.
Such as,
=(2104) (6102)
Then,
= (26) 104+2
So,
= 12108
Real-world applications of scientific notation in different fields are given below:
Chemists use scientific notation to measure or calculate the atomic size or the atomic level. Such as Avagardo’s number which can be written scientific notation 6.0221023 which can be approximately equal to the one mole in an atom or a molecule.
In space, astronomers used scientific notation to calculate the vast distance. Because in space the distance between the moon, the sun, and other planets is vast so that's why to calculate this distance in a short or efficient way astronomers used scientific notation. For example, the distance between the Earth and the nearest star is approximately equal to 4.241013 kilometers.
Engineers work on a large scale and calculate the calculation in a very precise and efficient way so that is why they use scientific notation to express large number calculations in short form. For example, the circumference of the earth which can calculated by engineers is 4.007107.
In physics, scientists can deal with very large and small quantities of measurements so that’s why to write these measurements in an efficient way they can use the scientific notation method. For example, the speed of light is 3.00108.
To become a scientist or for a scientific education understanding the scientific notation is essential. Scientific notation education starts in middle school but is explained in detail at higher levels. To become a scientist and, if we choose the engineering and technology career then the education of scientific notation is compulsory. because it helps them to calculate or, measure the very large or small measurements in a very precise or accurate form.
With the advantages and utility of scientific notation, it also becomes difficult or challenging for beginners. The common misconceptions that can be faced by beginners are given below:
Arithmetic operations rules can't be understood or can't be used precisely.
Zero can be confused with exponent.
The decimal point can't be placed properly.
Rules which are used for scientific notation can be misunderstood.
Strategies that can be used to manage or overcome these misconceptions or challenges include:
Understand the rules of scientific notation step by step and follow instructions.
Try to solve numerous examples without help.
Understand the rules and solve the arithmetic operations.
With the development of modern science and technology, the use of scientific notation is also increased and it can also be improved from time to time. In modern science and technology, very large and small measurements can be calculated so that’s why the scientific notation method is used to express calculation effectively and accurately because scientific notation also helps to overcome or analyze errors in calculations.
Examples of scientific notation are given below:
The mass of the electron is approximately equal to 0.000000000000000000000000000000910938356 kg
But we can write in scientific notation as,
= 9.10938356×10−31
The circumference of the earth is approximately equal to 40000000m
Then we can write in scientific notation as
= 4107
The distance between the sun and earth is approximately equal to 149,600,000 kilometers
In scientific notation, it can written as:
=1.496108
The mass of hydrogen is approximately equal to 0.00000000000000000000000167kg
In scientific notation, it can be written as,
=1.6710-27
The value 564300 can be converted into scientific notation and written as
=5.643105
Some problems are given below:
Convert the following problems into scientific notation:
456800
0.005544
98076
5544320
0.00000000009
9888800000
Solutions to these problems are given below:
4.568105
5.54410-3
9.8076104
5.544320106
910-11
9.8888109
Multiplication problems are given below:
(2108 ) (4104)
(2.0104) (2102)
Solutions to these problems are given below:
(24) 108+4
=81012
(2.02) 104+2
=4.0106
Division problems are given below:
5.61097.0103
8.41052.0102
A solution to these problems is given below:
8.0105
4.2103
Perform the following operations with numbers in scientific notation:
(5.0104)+(2.5104)
(6103) + (4.0103)
A solution to these problems is given below:
7.5104
6103
Hello friends, I hope you are all good. In our previous lecture, we discussed the SI Base Units in detail and today, we are going to discuss the units derived from these base units. In 1960, the International Committee conference was held and they presented the measurement units that are used to measure all quantities worldwide.SI units are used to make accurate and precise measurements.
SI units are the set of seven basic units called base units and all other units are derived from these base units and called derived units. A set of seven base units is used to measure the physical quantities but derived units are used to measure the complex quantities other than physical quantities. SI units play an essential role in modern technology and sciences.
The basic SI units from which the other units are derived are given below:
Units |
Symbol |
Measure |
meter |
m |
length |
Kelvin |
k |
Thermodynamic temperature |
Ampere |
A |
Electric current |
Second |
s |
Time |
candela |
cd |
Luminous intensity |
Kilogram |
kg |
mass |
mole |
mol |
Amount of a substance |
These units are the building blocks of all others known as derived units. Derived units are the combination of these base units in the form of an algebraic combination.
SI-derived units which are also known as coherent-derived units are derived from base units which can be expressed in ratio, product, or exponential form. Some derived units have dimensions but some do not because their demission cancels through their ratio or product.
Derived units are used to measure the complex quantity. Derived and coherent derived units have specific names, symbols, units, and dimensions. All derived units have dimensions except steradian or radian. 22 derived units, their names, symbols, and dimensions are given below:
Derived quantity |
Specific name |
symbol |
Specific symbol |
SI base units |
Other equivalent units |
Magnetic induction, magnetic flux density |
tesla |
T |
Jm |
Wb/m2 |
kg⋅s−2⋅A−1 |
Plane angle |
Radian |
α,𝛃,ℽ |
rad |
m/m |
— |
Solid angle |
Steradian |
Ω |
sr |
m2/m2 |
— |
Catalytic activity |
Katal |
ζ |
kat |
mol/s |
mols-1 |
Illuminance |
lux |
Ev |
lx |
lm/m2 |
lm.m-2 |
Absorbed dose |
Gray |
D |
Gy |
J/kg |
m2⋅s−2 |
Inductance |
henry |
L |
H |
Wb/A |
kg⋅m2⋅s−2⋅A−2 |
Pressure, stress |
pascal |
р |
Pa |
N/m2 |
kg⋅m−1⋅s−2 |
Electric resistance |
ohm |
R |
Ω |
V/A |
kg⋅m2⋅s−3⋅A−2 |
Celcius temperature |
Degree Celcius |
t,δ |
℃ |
k |
k |
force, weight |
Newton |
F |
N |
Kg m/s2 |
kg⋅m⋅s−2 |
Frequency |
hertz |
f,v |
Hz |
1/s |
s-1 |
Energy, work, heat |
Joule |
E |
J |
N/m , w/s |
kg⋅m2⋅s−2 |
Dose equivalent |
sievert |
H |
Sv |
J/kg |
m2⋅s−2 |
Luminous flux |
lumen |
Փv |
lm |
cd/sr |
cd |
Activity of radionucleotide |
becquerel |
A |
Bq |
1/s |
s−1 |
Magnetic flux |
weber |
Փ |
Wb |
V/s |
Vs−1 |
Capacitance |
farad |
C |
F |
C/V |
kg−1⋅m−2⋅s4⋅A2 |
Electric charge |
Coulomb |
Q,q |
C |
s/a |
s.a |
Electric potential difference |
Volt |
Vab |
V |
W/A |
kg⋅m2⋅s−3⋅A−1 |
Electric conductance |
Siemens |
B |
S |
A/V |
kg−1⋅m−2⋅s3⋅A2 |
SI units are used in different fields in chemistry, kinematics, thermodynamics, photometry, kinematics, electromagnetism, or in mechanics. The details are given below:
Derived quantity |
Name |
Symbol |
SI units |
Luminance energy |
Lumen second |
lm.s |
s.cd |
Luminance |
Candela per square meter |
cd/m2 |
m−2⋅cd |
Luminous exposure |
Lux second |
lx.s |
m−2⋅s⋅cd |
Luminous efficacy |
Lumen per watt |
lm/w |
m−2⋅kg−1⋅s3⋅cd |
Derived quantities |
Symbol |
Name |
Electric charge |
C |
Coulomb |
power |
W |
watt |
frequency |
Hz |
hertz |
Force |
N |
newton |
Voltage |
V |
volt |
area |
m2 |
Square meter |
Pressure |
Pa |
pascal |
Volume |
m3 |
Meter cube |
Energy |
J |
joule |
Derived quantities |
Names |
Symbol |
speed |
Meter per second |
m.s-1 |
Jounce |
Meter per fourth second |
m.s-4 |
Angular acceleration |
Radian per second square |
rad.s-2 |
Volumetric flow |
Cubic meter per second |
m3.s |
jolt |
Meter per cubic second |
m.s-3 |
acceleration |
Meter per second square |
m.s-2 |
snap |
Meter per fourth second |
m.s-4 |
Angular velocity |
Radian per second |
rad.s-1 |
Frequency drift |
Hertz per second |
hz.s-1 |
Derived quantities |
Names |
Symbols |
Thermal expansion |
Per kelvin |
k-1 |
Specific entropy |
Joule per kg per kelvin |
J.kg-1.k-1 |
Thermal resistance |
Kelvin per watt |
k.w-1 |
Heat capacity |
Jolue per kelvin |
J.k-1 |
Temperature gradient |
Kelvin per meter |
k.m-1 |
Thermal resistance |
Kelvin per watt |
k. w-1 |
Derived units play a very vital role in physics and many other fields for accurate and precise measurement and also help to understand the physical phenomena. The building blocks of SI units are seven basic units by combining them the all other units derived. Derived units play an essential role in modern science and technology to measure stable and precise measurements.
The importance of derived units in physics is given below:
Standardization and consistency
Efficiency in scientific computation
Practical applications
Simplification of complex quantities
Enhanced power solving
Facilitation of understanding and communication
Interdisciplinary relevance
Details of the importance of derived units are given below:
The use of derived units enhances computational efficacy in physics. Calculations involving derived units are often more straightforward than those using only base units. For instance, the
Unit watt for power simplifies calculation ]s involving energy and time, eliminating the need for repeated conversions from base units. This efficacy is particularly important in complex calculations, simulations, and models, where reducing the number of steps can significantly impact the accuracy and speed of results.
Derived units are not confined to physics alone but are used across various scientific disciplines, fostering interdisciplinary research and applications. For instance, the coulomb, a unit of electric charge, is crucial in both physics and chemistry. The use of common derived units across different fields facilitates collaboration and integration of knowledge, leading to achievements in areas such as materials science, biophysics, and environmental science. This interdisciplinary relevance underscores the versatility and universality of derived units in scientific inquiry.
One of the foremost reasons derived units are vital in their role in standardizing measurement globally. The SI system, established and maintained by the International Bureau of weights and Measures (BIPM), ensures that measurements are consistent and universally accepted. Derived units such as newton(N) for force, the joule (J )for energy, and the watt(W) for power, are constructed from base units like the kilogram(kg), meter(m), and second(s). This standardization is crucial for scientific communication, allowing researchers from different countries and disciplines to compare results and collaborate effectively on confusion or the need for conversion factors.
In practical and industrial contexts, derived units are designed to be more user-friendly for real-world measurements. For example, the hertz(Hz) is used to measure frequency, providing a more practical and comprehensible unit than cycles per second. Similarly. The use of derived units like the lumen(lm) for luminous flix or the sievert (Sv) for radiations enables more precise and accessible measurement and regulation in various industries, including healthcare, engineering, and environmental monitoring.
Derived units provide a clearer and more intuitive understanding of physical phenomena. For example, energy is measured in joules, which convey specific physical meaning, more straightforwardly than its base unit equivalent. This clarity extends to educational settings, where students can grasp complex concepts more easily when they are expressed in familiar derived units. additionally, using standardized derived units in scientific literature and presentations ensures that findings are communicated effectively, fostering a shared understanding among researchers.
Many physical quantities are inherently complex and can't be conveniently expressed using only base units. Derived units simplify these expressions, making them more manageable and intuitive. For instance, pressure is commonly expressed in pascals(Pa) rather than its base unit form of kg/ms-2. This simplification is not merely a matter of convenience but also aids in understanding and interpreting physical concepts more readily. It allows scientists and engineers to work more efficiently, reducing the potential for error and misinterpretation.
Derived units play a crucial role in problem-solving by aiding in dimensional analysis, which involves checking the consistency of equations. Dimensional analysis ensures that equations are dimensionally consistent, meaning that both sides of an equation have the same units. This technique is fundamental in verifying the correctness of quotations, deriving new relationships, and identifying potential errors. For example, ensuring that the units on both sides of Newton’s second law (F=ma) match confirms that the equation is dimensionally consistent and physically meaningful.
Some applications of derived units are given below:
This unit are used for heat, energy, and work. Their application in physics is given below:
Mechanics: the work which is done by a person or an object can be measured in joules. This concept is also used to understand the potential and the kinetic energy.
Thermodynamics: according to the first law of thermodynamics the energy that can be expressed in joules can't be created or can’t be destroyed and the energy in the form of heat can always be measured in joules.
Electricity: in the electrical system the energy which can be used in the form of electricity can be measured in joules because work is also done in it.
These units are used for power, their application in physics are given below:
Mechanical systems: the power that can be consumed in engines or different mechanical systems can be measured in watts.
Electrical engineering: in our daily life or households like bulbs, ovens, and other electronic devices, the electric energy that can be consumed is measured in watts.
Thermodynamics: In cars engines, and other electronic devices which consume electrical energy can be measured in watts.
These units are used to measure electric potential, and their application in physics is given below:
Power distribution: the voltages that can be distributed to the power line can be measured in volts.
Electronics: the electrical devices like capacitors, diodes, and others that can distribute the voltages. These voltages which can be distributed are measured in volts.
Electric circuits: in electric circuits, the potential that can be distributed is measured in volts.
These units are used to measure the magnetic flux density, their applications in physics are given below:
Physics research: In physics research when high magnetic fields are used then these field strengths can be measured in teslas.
Electromagnetism: the magnetic field strength in the magnets can be measured through teslas.
Medical imaging: In the medical field medical devices like MRI machines have also a strong magnetic field which can also be measured in teslas.
This unit is used to measure electric resistance, their applications in physics are given below:
Thermistor: temperature-sensitive resistors use temperature and resistance both and to measure or control the temperature ohm is used.
Circuit design: in electrical circuits where electric current is passed resistance is also present to measure the resistance we can use ohm.
Material science: to manufacture the electronic device, an ohm is used to measure the resistance.
This unit is used to measure inductance, their applications in physics are given below:
Radiofrequency: In radio when the tune is played then the frequencies are matched because during their manufacturing frequencies are measured in hertz.
Electrical engineering: To make inductors and coils of inductance to make a strong magnetic field Henry is used to measure the inductance.
Power supply: to supply the power current smoothly without changing in current and voltage inductors are used.
This unit is used to measure capacitance, their applications in physics are given below:
Energy storage: capacitors which are used in electrical devices are used to store energy and the farad is used to measure the capacitance of a capacitor.
Communication systems: In television, radios, and other electronic devices capacitors are used to store energy.
This unit is used to measure force, their applications in physics are given below:
Aerospace: Newton is used to find or determine the thrust and friction that are produced by engines during flight.
Classic mechanics: Newton is used to determine the forces that are produced by an object or a person. Newton also presented 3 laws.
Engineering: the engineers who build the buildings and bridges calculate the forces to ensure that these can bear stress or not.
This unit is used to measure pressure, applications of Pascal are given below:
Engineering: The engineer used Pascal to calculate the stresses of the material and then used the material.
Fluid dynamics: Blood pressure, atmospheric pressure, and all fluid pressure can be measured and calculated through Pascal.
Meteorology: to measure the climate and weather Pascal is used.
This unit is used to measure electrical charge, their application in physics are given below:
Capacitors: capacitors are used in electrical devices to store energy and to maintain the current flow.
Electrostatics: in the electrostatic experiments coulomb are used to measure the electric charge that flows through them.
Batteries: in batteries, the charge stored that is used after completing the charging.
Derived units are used in various fields to measure complex quantities and physical phenomena that can't be measured by using the base quantities. Derived units like joule, watt, kelvin, coulomb, Pascal, and all others are derived from basic seven base units in the form of exponent. Now derived units are used in modern technology and sciences. Scientists used these units to measure the different quantities precisely and accurately. From time to time, the SI system can also continue to adopt the changes and present more suitable and precise units for the measurement of the quantities. The SI system adopts many changes in its units and redefines them.
Hi, friends. I hope you are all well. Today we will discuss the SI ( International system of units) in detail. An international committee conference held in 1960 concurred on a set of definitions used to describe the physical quantities. This committee and the founded system are called System International (SI).
SI units are the measurement systems used generally to measure the standards.SI units play a vital role in measuring standard quantities in scientific and technological research.SI units are a set of basic 7 base units from which derived units are defined.SI units also play a vital role in modern metrology and now they become a part of the foundation of modern science and technology.SI units can be categorized into three types:
Now we will discuss what SI base units are, their definitions, importance, and applications in detail.
SI base units are the basic standard units explained by an international system (SI) of units. SI base units are known as the building block of the international system of units because all other units are derived from these basic standard units.
Base units are used to express the base quantities. And the other units are used to express the derived quantities which are derived from base quantities and units. The physical quantity and units are:
These SI units were globally accepted for measurement of the physical quantities.
The basic definitions of the SI base units are given by the System international unit in detail below:
Unit |
Measure |
Symbol |
Typical symbols |
Definition |
meter |
length |
m |
L,x,r, etc |
In the SI unit system the, meter is the unit of length and it is defined as in 1/299, 792,458 seconds the light travels in a vacuum. In 1983 this definition was presented and it is based on the fundamental constant of nature, the speed of light. meter is the most precise unit which are used in the measurement. |
second |
Time |
s |
t |
The unit of time is second and it is defined as the transition changing between the two levels of hyperfine at the ground state of the cesium atom. This definition was presented in 1967 and also defined as 9,192,631,770 duration of radiation in between the transition. Now to measure the accurate time the unit second is used in science. |
Kilogram |
mass |
kg |
m |
SI of mass in kilogram it is defined by IPK as the international prototype of kilogram and in simple words, it is defined as the mass of a substance in thousand grams. Now this unit is widely used in measurements of the mass of an object. |
Ampere |
Electric current |
A |
I, i |
The unit of the electric current is ampere. The definition of unit electric current is based on the charge of a proton and the elementary charge e, and conductor forces. But with time it can change and be explained on the fixed numerical value of elementary charge. Now the precise and accurate measurement of electric current in the unit ampere can be used. |
kelvin |
Thermodynamic temperature |
k |
T |
The unit of thermodynamic temperature is kelvin. It was defined based on the Boltzmann constant(k) and also relates to the average kinetic energy of the gas. but now it can be defined on the basis of a fixed numerical value of the Boltzmann constant to measure the accurate temperature with the help of their unit Kelvin. |
candela |
Luminous intensity |
cd |
lv |
The SI unit of luminous intensity is candela and it is used to measure the power of light. It is defined as measuring the radiation of the frequency of 540x10hertz which is emitted from monochromatic sources and also measures the radiant intensity of 1/683 watt per steradian. Now to measure the accurate and exact power of light candela unit is used. |
mole |
Amount of a substance |
mol |
n |
The SI unit of the amount of a substance is mole.it is defined as atoms or molecules in the carbon-12 isotopes, which is based on Avagord’s number 6.022x10 power 23. But now it is based on the fixed numerical value of Avagord’s number to measure the accurate amount of the substance with the unit mole. |
These definitions are the old and basic definitions but with time and with more research these definitions can be revised and new definitions of these base units are presented.
In 2016 November 16 the old and basic definitions of base units can be revised but it can be effective from 2019 May 20. The redefinition of the meter can be revised by understanding the physical artifact and it is not based on the property of nature. Other basic units like candela, kilogram, mole, and ampere can be connected to the revised definition of kilogram which is presented by an international prototype of the kilogram, by storing the cylinder of platinum-iridium in a vault near Paris.
The revised and new definitions of SI base units are given below.
Units |
measures |
Dimensions |
Symbol |
Revised and new definitions |
Origin |
Candela |
Luminous intensity |
J |
cd |
The SI unit of luminous intensity and the power of light is candela, symbol cd. It is stated and based on the fixed numerical value of luminous efficacy of 540×1012 Hz frequency of monochromatic radiation. It can also expressed in W−1 or also in kg−1 m−2 s3 |
The principle and the base of the candle power are the standard properties of burning candles which can emit light traditions through burning. |
Ampere |
Electric current |
I |
A |
The SI unit of electric current is the ampere, symbol A. it can based and taken from the fixed numerical value of e (elementary charge) which is equal to 1.602176634×10−19 and also expressed in units C and s. |
Specifically the unit ampere at internationally defined and based on the electrochemical. In the electrochemical process, the current is required to store 1.118 mg of silver per second to form the solution of silver nitrate. |
Mole |
Amount of a substance |
N |
mol |
The SI unit of the amount of a substance is a mole and the symbol is mol. Mole is based on Avagord’s numbers so one mole is equal to 6.022 140 76 × 1023 And this value is the fixed numerical value of Avagord’s number which can’t be changed. And their unit is per mol. To express the substance the symbol which is used is n. N is for the elementary entities that can be specified and the elementary entities may be an atom, molecule, ion or electron, or a group of particles. |
1 g/mol is equal to the molecular weight which is divided by the molar mass. |
meter |
Length |
L |
m |
The SI unit of length is meter, symbol is m. it is defined on the principle of the speed of light in vacuum c which is equal to 299792458 and expressed in unit ms-1. |
The median arc through Paris is measured and the total distance which is measured is equal to 1/100000000 from Earth to the north pole. |
Kilogram |
mass |
M |
kg |
The SI unit of mass is the kilogram, the symbol is kg. It is based on the fixed numerical value of the Planck constant which is represented by h and equal to 6.62607015×10−34 and their unit is Js. |
One liter is equal to thousands of m3. The mass of water is equal to the temperature of melting ice. |
Kelvin |
Thermodynamic temperature |
⊝ |
k |
The SI unit of thermodynamic temperature is kelvin, symbol is k. It can also defined based on the Boltzmann constant the fixed numerical value which is equal to 1.380649×10−23 And their unit is JK-1. |
The Kelvin and the Celsius scale both are used in thermodynamicscic temperature and 0k is equal to the absolute zero. |
Second |
Time |
T |
s |
The SI unit of time is second, symbol s. The fixed numerical value of cesium defined it. Because the ground state hyperfine transition frequency of cesium is equal to 9192631770 and its unit is hertz which is expressed in s-1. |
Each day has 24 hours and each hours have 60 seconds and 1 second is equal to the 24x60x60 of each day. |
Application and importance of all seven base units in detail are given below:
Physics and engineering: to measure the accurate length in constructing the building and in designing the machinery.
Astronomy: to measure the distance between the objects and the measurement is precise and accurate.
Everyday life: in our daily life meter is used to measure the length of an object and also measure the distance or the plot size.
Lighting industry: to measure the precise and accurate luminous intensity which is important in designing and creating a lighting system.
Cinematography and photography: to achieve the effective and desired visual effects accurate measurement of light is essential.
Vision science: to understand or design human vision aids measurement of luminous intensity is very essential.
Synchronization: accurate and precise measurement of time is very important for clocks at the global level and also shows an impact on telecommunication and in many other social systems.
Daily life: in daily life time measurement is essential to make a schedule and to do work on their given time.
Physics experiment: to measure the velocity, speed, and distance time measurement is essential to do work or experiment on time.
Medical field: to make the dosages and the multivitamins the mass of a substance is essential to measure because if the mass is increased to make a low dose of medicine then it can show very adverse effects.
Science and industry: to make the chemicals in industry accurate and precise measurement of a substance is very important.
Trade and commerce: for fair trading the measurement of mass is essential.
Environmental science: in the environment, the amount of pollutants and harmful gases can be measured through a mole.
Chemistry: In the chemical reaction of chemistry the mole is the essential and fundamental part of a reaction or the chemistry experiment
Pharmacology: in the pharmaceutical industry the precise and accurate measurement of mole is essential to make chemicals and medicines.
Medical equipment: many medical machines like x-ray machines, MRIs, and CT scans can measure the precise measurement of electrical current.
Electrical engineering: to measure the precise and accurate current in electronic devices.
Power system: to distribute the electric current in the electrical system and to operate the power system the measurement of electric current is essential.
Industrial process: in industries to manufacture the material the accurate and precise measurement of temperature is vital.
Climate studies: the global and climate temperature change every time and to measure the accurate temperature of climate, the temperature can be measured accurately.
Scientific research: in physics, chemistry, and many other science subjects in which we can perform experiments, the measurement of temperature is essential to run the process of experiment.
The foundation of all SI units in the SI system are the seven base units which are interlinked. To measure the complex and difficult physical quantities derived units are used which are a combination of seven base quantities. Some examples are given below:
Watt(W): this is the unit of power and can derived from the base unit (J/s).
Newton(N): this unit is used for force and derived from the base unit meter, seconds, and kilogram.
Joule(J): this unit is used for energy and derived from base units like seconds, kilograms, and meters.
These are some examples of derived units that are derived from the combination of different base quantities.
The SI unit's definition can be changed with time to increase precision and stability. because these units can be used in modern science and technology and also in meteorology processes.
The SI system can do more research to evolve modern technology and discover more precise and effective results of their research to make the bright future of SI units. Potential future changes could involve redefining units based on even more fundamental principles or developing new measurement techniques that further enhance precision and accessibility.
The international system of units(SI) plays a vital role in physics, scientific research, pharmaceutical industries, and our daily lives. The seven base units are the fundamental and the foundation of the international system of units. These units help to measure all the quantities in a very effective and the results are precise and accurate. With time science and technology progress the si system also tries to maintain more precise and relevant changes in their research.
I will let you guys know about how velocity is a regular part of our daily lives and how it behaves in the environment we are living in. To understand the basic concept we need to have a deeper look at its real-life examples. A detailed discussion on velocity to have a better understanding is provided in the next section. Let’s get started.
An earthly object can possibly have two states i.e. rest or motion. If an object is in motion, a numerical value called Speed is used to measure how fast or slow the object is moving? Speed is defined as the distance covered per unit of time. So, if an object covers a distance of 1 meter in 1 second, its speed will be 1m/s. As speed is a scalar quantity so it just gives the scalar information(about motion) and doesn't tell us anything about the direction of the movement i.e. object is moving towards north, south or may have a circular motion.
So, in order to completely define the motion of an object, an equivalent vector quantity of speed was introduced and named Velocity. Velocity, not only gives the numerical value(speed) but also tells the direction of the moving object. In simple words, speed plus direction is equal to velocity and as speed is distance per unit time, similarly velocity is displacement per unit time.
Now let's have a look at a proper definition of Velocity:
Let's have a look at the symbol of velocity:
Now let's have a look at the mathematical formula for calculating the velocity of an object:
Velocity = Displacement / Time
v = d/t
As v & d are both vector quantities, so written in bold while t is a scalar quantity.
Average Velocity = Distance Covered / Total Time
?v = ?d/?t
?v = (d2 - d1) / (t2 - t1)
where t1 & t2 are initial and final time intervals and d1 and d2 are initial and final displacements of the object.
Now, let's drive the velocity unit from its formula:
Velocity = Displacement / Time
where SI unit of displacement is the meter and that of time in seconds.
Velocity = meter / second
In the game of cricket, the velocity of the ball is usually not measured in SI units rather they measure it in either kilometer per hour or miles per hour.
Velocity Dimension = [L/T]
v = [LT-1]
Depending upon various factors, velocity has been divided into multiple types as discussed below. Let’s read through them all.
Let's understand it with an example of a ball thrown upwards:
As we know, Earth's gravitational force pulls everything towards it. So, considering the earth as a reference point, when you throw a ball in the upward direction, it's moving away from its reference point(Earth's center). So, during its upward flight, the ball will have a negative velocity and thus is written with a negative sign.
Let's continue that example of the ball moving upward:
As we have seen in the previous section, the ball will have a negative velocity while moving upward. But when it will reach the maximum height and rite before moving back in the downward direction, for an instance it will have a zero velocity, as it won't be moving either upward or downward.
Let's add some more in that ball example:
Once the ball reaches the maximum height, it will start moving back in the downward direction. Now, the ball is moving towards its reference point(Earth's Core) so it will be said to have positive velocity now.
Let's understand it with the same example:
We have seen the ball example thrown upward. If we consider both of its loops(moving upward and then downward), its initial velocity will be right where it left the hand of the thrower. It will have a maximum initial velocity as during the upward direction it will slow down and during the downward direction, it will lose some to friction. But if we only consider the second loop i.e. the ball has reached its maximum position and now it's moving downwards. So, in this scenario, the initial velocity of the ball will be 0. I hope it got cleared.
v = u + at
u = v - at
The above expression shows when we multiply acceleration with the given time and subtract this product from the final velocity, it gives us the initial velocity.
u2 = v2 - 2aS
u = S/t - (1/2) at
u = 2(S/t) - v
where,
v = u + at
orVf = Vi + at
Vf2 = Vi2 + 2aS
Where,Let's understand the concept associated with the final velocity through a visual example.
A projectile motion of the ball thrown from one end is shown in the figure below. At time zero (t = 0), when a guy in a purple shirt throws a ball, the velocity of that ball at this time is considered initial velocity. After reaching a particular height, when the ball starts moving downwards and reaches at t = 8 seconds in the hands of a guy wearing a green shirt. At t = 8 seconds, the velocity of the ball is the final velocity. After this velocity, an object comes again into the stationary position.
Similarly, if you drop a ball from a specific height and allow it to move towards the ground as shown in the figure below. The moment you drop the ball, the velocity is called initial velocity. Whereas, the moment when the ball touches the ground, the velocity will be known as the final velocity.
Now let's have a look at different types of velocity in detail:
Depending on the type of object and its motion, we have numerous types of velocities, a few of them as discussed as follows:
Average velocity = total displacement covered / total time taken
?v=?x/?t
?v = (x2-x1) / (t2-t1)
Where,Average velocity cannot tell us how fast or slow an object is moving in a specific interval of time and for that, we have another type of velocity called Instantaneous velocity.
By applying a limit “t” approaches zero on the average velocity provides us with the instantaneous velocity as shown in the formula given below.
Vinst = Lim t -> 0 (?d/?t)
Take a look at the figure below, the velocity at point “p” depicts the instantaneous velocity of a moving body.
The figure below shows the relation between average and instantaneous velocity. The velocity is represented by the red line and has been divided into several segments. The position is displayed on the y-axis whereas the x-axis shows the time consumed. In the first interval, Jack has covered 3 miles in the first 6 minutes. In the second interval, Jack stopped for 9 minutes. Whereas, in the third interval, Jack covered another 5 miles in 15 minutes. If we divide the total displacement covered by Jack by the total time consumed during the whole travel, it will give us an average velocity.
x=xo+vt
Where,xo=position of the body at t=0
a=dv/dt=0 v=constant
This scenario can be visualized through a velocity-time graph as shown in the figure below. You can see a straight line for each time interval depicting the velocity is constant throughout with “0” acceleration.
Let's understand this from a real-life example.
For instance, if a fan installed in your room is rotating at a continuous speed, its velocity will be variable because its direction gets changed every time.
Vorbit=GMR
Where,ve=2GMr
Where,Let's have a look at how to find the angular velocity of a moving object?
Angular Velocity FormulaTo calculate this quantity, a formula is given below.
?=??/?t
Or,
?=v/r
Where,
The direction of motion of an object moving with angular velocity is always perpendicular to a plane of rotation. It can be measured using the right-hand rule. The whole concept is shown in the figure below.
The above figure shows that the linear velocity is dependent on the two different parameters i.e., distance covered and the time consumed to cover that particular distance.
Let's have a look at how to find linear velocity?
Linear Velocity FormulaIt can be calculated using the below mathematical expression.
velocity=distance/time
v=S/t
As we know,
S=r?
Putting this value in the above formula we have,
v=r?/t
The linear velocity can also be represented in terms of an angular velocity as given below.
v=r?
vt=2mgACd
Where,
Let's understand with an example.
Let's understand this with a visual example.
The track of a car moving with non-uniform velocity is shown in the below figure. Unequal displacements covered in equal intervals of time can clearly be seen from the velocity-time graph.
Let's understand the overall scenario with an example.
For instance, the air is causing some hindrance in the airplane’s track or a boat is traveling through the river whose water is flowing at a particular rate. In such cases, to observe the complete motion of the object, we need to consider the effect of the medium affecting the motion of a moving body. By doing so, we measure the relative velocity of that moving object as well as the medium’s velocity affecting its motion
Let's have a look at another example to have a better understanding of relative velocity.
Finding Relative VelocityVxy=Vx-Vy
Vyx=Vy-Vx
Vxy=-Vyx
|Vxy|=|Vyx|
It has been proved through various research studies that most of the time people get confused when it comes to speed and velocity. They mostly get confused in implementing their concepts separately in different scenarios as and when needed.
If I tell you the very basic difference between these two quantities, they are just as different as distance and displacements are.
Let's have a look at some more points to understand the difference effectively.
Therefore, keeping in mind the above points, it can be said that a direction creates a major difference between speed and velocity.
Let's understand through an example.
For instance, 30 kilometers per hour is the speed of a moving vehicle whereas 30 kilometers per hour east shows the velocity of the same vehicle.
| Parameters | Speed | Velocity |
| Definition | The rate at which a body covers a particular distance is commonly known as speed. | The rate at which a body changes its position in a specific direction is called velocity. |
| Magnitude | Speed is always positive and it cannot be either negative or zero. | Velocity can be positive, zero, and negative depending upon the direction in which an object is moving. |
| Quantity Type | Speed does not need any direction for its description so, it is a scalar quantity. | Velocity cannot be described without direction so it is a vector quantity. |
| Change in Direction | Change in direction does not matter when calculating average speed. | Every change in direction changes the velocity. |
| Formula | s=distancetime=dt | s=change in positionchange in time=st |
| SI Units | Meter per second (m/s) | Meter per second (m/s) |
A few examples of velocity from real-life are presented to clear your concepts related to it if there still exists any confusion.
This is all from today’s article. I have tried my level best to explain to you each and everything associated with the velocity. I have focused in detail on its basic concept, various forms, unit assigned by System International, and visual examples where needed. Moreover, I have provided you with a couple of examples captured from real life so that you can have a better understanding of velocity.
I hope you have enjoyed the content and are well aware of this topic now. If you are looking for more similar information, stay tuned because I have a lot more to share with you guys in the upcoming days. In case you have any concerns, you can ask me in the comments. I will surely try to help you out as much as I can. For now, I am signing off. Take good care of yourself and stay blessed always.
Thank You!
| Greatest Physics Scientists(Physicists) of all Times | ||||
|---|---|---|---|---|
| No. | Physicist Name | Achievement | ||
| 1 | Albert Einstein | Theory of Relativity(E=mc2), Quantum light theory, Avogadro's Number etc. | ||
| 2 | Isaac Newton | Laws of motion, Gravitational force etc. | ||
| 3 | Nikola Tesla | Worked mostly in Electrical Energy. | ||
| 4 | William Gilbert | Proposed that Earth is a giant magnet. | ||
| 5 | Willbrod Snell | Laws of refraction i.e. Snell's Laws. | ||
| 6 | Galileo Galilei | worked in astrophysics. | ||
| 7 | Blaise Pascal | Famous for Pascal's Laws. | ||
| 8 | Daniel Bernoulli | Famous for Bernoulli's Theorem. (Fluid Flow) | ||
| 9 | Christiaan | Famous for Hagen’s Principle. (Geometrical Theory for Light) | ||
| 10 | Benjamin Franklin | He discovered electrical charges. | ||
| 11 | Leonard Euler | Worked in Fluid dynamics, Lunar Theory, Mechanics etc. | ||
| 12 | Henry Cavendish | Worked in GeoPhysics. | ||
| 12 | Joseph Louis | Worked in Mechanics. | ||
| 12 | Robert Milikan | Worked on charges & cosmic rays etc. | ||
| 12 | C. Wilson | worked in GeoPhysics. | ||