Dimensional Formula for Work and Energy Explained

Understanding the dimensional formula for physical quantities like Work and Energy is fundamental in physics. Let's derive and explain the correct dimensional formula step-by-step.

Defining Work in Physics

Work is a concept in physics that describes the energy transferred when a force acts upon an object, causing a displacement. It is calculated by multiplying the force applied to the object by the distance the object moves in the direction of the force.

The basic formula is:

Work = Force × Distance

We denote the dimensions of Work as [W].

Deriving Dimensions of Force

To find the dimensions of Work, we first need the dimensions of Force.

According to Newton's second law of motion:

Force = Mass × Acceleration

Let's determine the dimensions for Mass and Acceleration:

• The dimension of Mass is represented by [M].
• Acceleration is the rate of change of velocity (change in speed over time). Velocity is distance over time.
  • Dimension of Velocity: $\frac{\text{Distance}}{\text{Time}} = \frac{[L]}{[T]} = [LT^{-1}]$
  • Dimension of Acceleration: $\frac{\text{Velocity}}{\text{Time}} = \frac{[LT^{-1}]}{[T]} = [LT^{-2}]$

Now, we can find the dimensions of Force ([F]):

[F] = [Mass] × [Acceleration] = [M] × [LT^-2]

So, the dimensional formula for Force is $[MLT^{-2}]$.

Calculating Dimensions of Work

Using the definition Work = Force × Distance, we can now calculate the dimensions of Work ([W]).

We know:

• Dimensions of Force ([F]): $[MLT^{-2}]$
• Dimensions of Distance ([d]): $[L]$

Therefore:

[W] = [F] × [d] = $[MLT^{-2}] \times [L]$

Combining the dimensions:

[W] = $[ML^{1+1}T^{-2}] = [ML^2T^{-2}]$

The dimensional formula for Work is $[ML^2T^{-2}]$.

Energy and Dimensional Consistency

Energy is defined as the capacity to do work. In physics, all forms of energy (like kinetic energy, potential energy, thermal energy, etc.) are equivalent to work in terms of their dimensions. This means they share the same dimensional formula.

For instance:

• Kinetic Energy (KE) depends on mass and velocity ($KE = \frac{1}{2}mv^2$). Its dimensions are $[M][LT^{-1}]^2$, which simplifies to $[ML^2T^{-2}]$.
• Gravitational Potential Energy (PE) depends on mass, gravity, and height ($PE = mgh$). Its dimensions are $[M][LT^{-2}][L]$, which also simplifies to $[ML^2T^{-2}]$.

This confirms that the dimensional formula for Energy ([E]) is also $[ML^2T^{-2}]$.

Identifying the Correct Dimensional Formula Option

Let's compare the derived dimensional formula $[ML^2T^{-2}]$ with the options provided:

• Option 1: $[MLT^{-2}]$ represents the dimensions of Force.
• Option 2: $[MLT^{-1}]$ represents the dimensions of Linear Momentum.
• Option 3: $[ML^2T^{-2}]$ correctly represents the dimensions of Work and Energy.
• Option 4: $[ML^2T^{-3}]$ represents the dimensions of Power (Work done per unit time).

Therefore, the correct dimensional formula for Work or Energy is $[ML^2T^{-2}]$.