Dimensions of physical quantities

The Dimensions of physical quantities define its nature. When we see an item, the dimensions are the first thing we notice. Indeed, we are characterised or observed in our dimensions, i.e., our height, weight and amount of flesh. The dimension of a body refers to its relation to fundamental quantities. When we define a quantity’s dimension, we are essentially defining its identity and existence. It becomes obvious that everything in the cosmos has a dimension and so a presence.

A Dimension of physical quantities is defined as the powers to which fundamental quantities are elevated to express it. The dimensions of the seven fundamental quantities are denoted by square brackets []. 

Examples 

The dimension of length is denoted by [L], the dimension of time by [T], the dimension of mass by [M], the dimension of electric current by [A] and the dimension of quantity by [mol]. 

The temperature dimension is [K], while the luminous intensity dimension is [Cd]. 

Consider a physical quantity Q dependent on fundamental quantities such as length, mass, time, electric current, substance concentration and temperature when increased to powers a, b, c, d, e and f. Then, the dimensions of physical quantities Q may be expressed as follows: 

[Q] is equal to [LaMbTcAdmoleKf]. 

It is required that we utilise [] when writing the dimension of a physical quantity. In reality, everything is expressed in mass, length, and time. Consider the following examples: 

1. A solid’s volume is equal to the product of its length, width and height. Its dimensions are as follows: 

• Volume equals the sum of the length, breadth and height. 

• Volume equals [L] [L] [L] (as length, breadth and height are lengths) 

• Volume equals [L3] 

• Because volume is mass and time-dependent, the powers of time and mass used to represent its dimensions, i.e. [M0] and [T0], will be zero. 

• The volume’s ultimate dimension will be [M0][L3][T0] = [M0L3T0]. 

2. Similarly, the area dimensions will be [M0][L2][T0]. 

3. The speed of an item is defined as the distance it travels in a given amount of time and is expressed as: 

• Distance/Time = Speed 

• Distance Dimension = [L] 

• Time Dimension = [T] 

• Speed Dimension = [L]/[T] 

• [Velocity] = [L][T-1] = [LT-1] = [M0LT-1]. 

4. Acceleration is defined as the rate at which a body’s velocity changes with respect to time. Its dimensions are as follows: 

• Acceleration is defined as Velocity / Time. 

• Velocity dimension = [LT-1] 

• Time dimension = [T] 

• The acceleration dimension will equal [LT-1]/[T]. 

• [Acceleration] = [LT-2] = [M0LT-2]

5. A body’s density is defined as its mass per unit volume, and its dimensions are as follows: 

Density equals mass divided by volume. 

Mass dimension = [M] 

Volume dimension = [L3] 

The density dimension will be equal to [M] / [L3]. 

[Density] equals either [ML-3] or [ML-3T0]. 

6. Force exerted on a body is the product of the body’s acceleration and mass. 

Acceleration = Force = Mass 

Mass Dimension = [M] 

Acceleration Dimension = [LT-2] 

Force dimension will be equal to  [M] × [LT-2]

Quantities Possessing the Identical Dimensional Formula :

• Momentum and impulse. 

• Work, torque, the moment of force and energy are all terms that refer to the same thing. 

• Rotational impulse, angular momentum, Planck’s constant. 

• Stress, pressure, elasticity modulus, and energy density. 

• Surface energy, surface tension, and force constant. 

• Angular velocity, frequency, and gradient of motion. 

• Potential gravitational energy, latent heat. 

• Thermal capacity, entropy, universal gas and Boltzmann’s constant are all constants. 

• Force, thrust.

• Power, luminous flux

The Advantages of Dimensions 

• Before beginning to write measurements for a physical quantity, it is necessary to understand why dimensions are necessary and what the benefits of describing a physical quantity are. The following are the advantages of describing a physical quantity: 

• Describing dimensions aids in comprehending the relationship between physical quantities and their dependency on basic or base quantities, i.e., how the dimensions of a body are determined by mass, time, length and temperature. 

• Dimensions are utilised in dimension analysis because they allow for the Unit Conversion and exchange of units. 

• Dimensions are utilised to forecast unknown equations by examining how a particular body is dependent on fundamental quantities and to what extent. 

• It simplifies the measurement and analysis of physical quantities. 

• We may identify or notice a quantity solely on the basis of its size. 

• Dimensions encapsulate the essence of items and their existence. 

Dimensional Analysis Explained Limitations 

• This approach cannot be used to determine dimensionless quantities. This approach cannot be used to find the constant of proportionality. They can be discovered either experimentally or theoretically. 

• This procedure does not work with trigonometric, logarithmic or exponential functions. 

• This strategy will be difficult to apply to physical quantities reliant on more than three physical qualities. 

• In some instances, the constant of proportionality has dimensions as well. In such instances, we are unable to use this system. 

• We cannot use this procedure to obtain an expression of one side of the equation that comprises the addition operation of physical quantities.

Conclusion

In this chapter, you have read about Dimensions of physical quantities containing various types of formulas that even includes dimensional formulas, the system of units, CGS, MKS, etc., system, Basic units and derived units. Dimensional analysis is an incredible technique for determining the dimensions of physical quantities to verify their relationships. The homogeneity principle is founded on the concept that two quantities of the same dimension may only be added, subtracted or compared. When the quantities involved are known, the dimensional analysis may be utilised to build credible equations.