Units and Measurements-Dimensional Formula

The topic ‘Units and measurement’ deals with the quantities used as a standard of measurements. It explains the comparison and measurement of certain physical quantities. We need some fixed quantities as standard units for the comparison and measurement. For example, an elephant’s weight is more than a horse’s, but how do we ascertain this? Rahul is taller than rony, but how tall is he? To find answers to such questions and make comparisons and measurements, we need standard units for different quantities. For example, length, weight, time, capacity, etc., have their standard units for measurement.

Dimensions in units and measurements

The dimensions can be written as the powers of the fundamental units of length, mass, and time. It depicts their nature and does not show their magnitude. 

Example to write dimensions

Let’s take the formula of the area of the rectangle:

Area of the rectangle = length x breadth

= Lx L ( where breadth is also showing the length of the side)

= [L1] X [L1]

= [L2]

Here, we can see the length to the power of 2 and we cannot find the dimension of mass and time.

Hence, the dimension of the area of a rectangle is written as [M0 L2 T0]

Dimensional formula dimensional equation

The dimensional formula depicts the dependency of physical quantity with fundamental physical quantity, along with the powers.

Example

Let’s take the formula of speed

Speed = Distance / Time

The distance can be written in length [L]

Time can be written as [T]

The dimensional formula would be [ M0 L1 T-1]

Hence, we can conclude that the speed is dependent on only length and time, not mass.

Dimensional equation

The physical quantity is equated with the dimensional formula, to get the dimensional equation. 

Example

Velocity =  [ M0 L1 T-1]

Here, velocity is the physical quantity, which is equated to the dimensional formula.

Here is the list of various physical quantities and their dimensions.

Physical quantity

In physics, we can find many physical quantities classified with respect to dimensional analysis.

On basis of a constant:

• Dimensional constant,

The dimensional constant has both fixed value and dimension value.

Eg: Gas constant.

• Dimensionless constant

The dimensionless constant has only a fixed value without dimension value.

Eg: the mathematical constant .

On basis of variable:

• Dimensional variable

The dimensional variable has a dimension value, not a fixed value.

Eg: Force

• Dimensionless variable

The dimensionless variable has both fixed value and dimension value.

Eg: Angle

Characteristics of dimensional formula and equation

The dimensional formula and principle are based on the principle of homogeneity of dimensions. The principle can be used when all the physical quantities are of the same nature. All the dimensions used in physical quantities should have the same dimensions. These are the most important characteristics of Dimensional Formula & Equations. 

This principle checks the correctness of the physical equation. For example, if we are having the power of 2 for all the terms like  L, M, T on the left side. The right side terms should also have the same power. Then, we can confirm that the physical equation is correct.

Illustration

Let’s take the equation, S = ut + ½ at2

Here. S is the length

U is the velocity

T is the time.

A is the acceleration.

The length is equated to velocity and time. Let’s see the dimensions of each term given.

For S, the dimension would be  [L1]

For u, the dimension would be [ L1T-1]

For t, the dimension would be [ T-1]

For a, the dimension would be [ L1T-2]

½ is a dimensionless constant in the physical equation. So, we will consider it in expression.

Comparison

LHS

The dimension of S is [ L1]

RHS

ut + ½ at2 = [ L1 T-1 ] [ T1] + [ L1T-2 ] [ T2]

= [ L1] + [ L1]

LHS = RHS =[ L1]

Through the dimensional analysis, we found that the terms on both sides are equal. So, the given physical equation is dimensionally homogeneous.

These are the most important characteristics of Dimensional Formula & Equations. 

Dimensional analysis

In a physical relation, the dimensions are examined through dimensional analysis. These analyses can be used in conversion, correction, and derivation.

Applications of dimensional analysis

It determines the dimensional consistency, homogeneity, and accuracy of the mathematical expressions. 

Limitations of the dimensional equation

• The principle of homogeneity of dimensions cannot be used for trigonometric and exponential expressions. The derivation is more complex and complicated.
• The comparing terms or factors are less.
• The correctness of the physical expressions depends only on dimensional equality.
• It is majorly used in the case of dimensional constant. We are not able to find the value of the dimensionless constant. 

Conclusion

In this topic, units of measurement, we have learned that we need fixed standard units to do measurements and comparisons. These standard units are accepted worldwide. The body named ‘General Conference on Weights and Measures’ has the right to decide and mention units. There are two types of quantities: fundamental and derived quantity. There are seven fundamental quantities: kilogram, second, kelvin, ampere, candela, meter, and mole.

Derived quantities are those which depend on the other quantities for their measurements. To study units of measurement,  we need to understand the topic of SI unit prefixes. There are 20 SI unit prefixes that are used to form decimal multiples and submultiples of SI units.