Question

The quantity which has the same dimensions as that of gravitational potential is

• A. latent heat
• B. impulse
• C. angular acceleration
  (D) specific heat capacity
• D. Planck's constant
• E.

Hint:

Dimensional analysis can often be simplified by looking at the definitions. Both gravitational potential and latent heat are defined as "energy per unit mass". This relationship immediately implies they must have the same dimensions.

Answer 1

The Correct Option is A

Step 1: Determine the dimensions of Gravitational Potential.
Gravitational potential (\( V_g \)) at a point is defined as the work done per unit mass in bringing a test mass from infinity to that point. \[ V_g = \frac{\text{Work Done}}{\text{Mass}} = \frac{W}{m} \] The dimension of Work (or Energy) is \( [W] = \text{ML}^2\text{T}^{-2} \). The dimension of Mass is \( [m] = \text{M} \). Therefore, the dimension of gravitational potential is: \[ [V_g] = \frac{[W]}{[m]} = \frac{\text{ML}^2\text{T}^{-2}}{\text{M}} = \text{L}^2\text{T}^{-2} \]

Step 2: Determine the dimensions of each of the given options.
(A) Latent Heat (L): It is the heat energy absorbed or released per unit mass during a phase change. \[ L = \frac{\text{Heat Energy}}{\text{Mass}} = \frac{Q}{m} \] The dimension of Heat (Energy) is \( [Q] = \text{ML}^2\text{T}^{-2} \). \[ [L] = \frac{\text{ML}^2\text{T}^{-2}}{\text{M}} = \text{L}^2\text{T}^{-2} \] This matches the dimensions of gravitational potential. (B) Impulse (J): It is the change in momentum, or Force × Time. \[ [J] = [\text{Force}] \times [\text{Time}] = (\text{MLT}^{-2})(\text{T}) = \text{MLT}^{-1} \] (C) Angular Acceleration (\(\alpha\)): It is the rate of change of angular velocity. \[ [\alpha] = \frac{[\text{Angular Velocity}]}{[\text{Time}]} = \frac{\text{T}^{-1}}{\text{T}} = \text{T}^{-2} \] (D) Specific Heat Capacity (c): It is the heat energy required per unit mass per unit change in temperature. \[ [c] = \frac{[\text{Energy}]}{[\text{Mass}][\text{Temperature}]} = \frac{\text{ML}^2\text{T}^{-2}}{\text{M}\Theta} = \text{L}^2\text{T}^{-2}\Theta^{-1} \] (E) Planck's Constant (h): From \( E = hf \), it is Energy per unit frequency. \[ [h] = \frac{[\text{Energy}]}{[\text{Frequency}]} = \frac{\text{ML}^2\text{T}^{-2}}{\text{T}^{-1}} = \text{ML}^2\text{T}^{-1} \]

Step 3: Compare the dimensions and conclude.
By comparing the dimensions calculated in Step 2 with the dimension of gravitational potential (\( \text{L}^2\text{T}^{-2} \)), we find that only latent heat has the same dimensions.

Final Answer: (A) latent heat