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Physics Question on Dimensional Analysis

If G be the gravitational constant and u be thee nergy density then which of the following quantity have the dimension as that the UG\sqrt{UG}

A

Pressure gradient per unit mass

B

Force per unit mass

C

Gravitational potential

D

Energy per unit mass

Answer

Force per unit mass

Explanation

Solution

Dimensional analysis:

[uG]=(M1L−1T−2)×(M−1L2T−3)=[M2L2T−4][uG] = \left( M^1L^{-1}T^{-2} \right) \times \left( M^{-1}L^2T^{-3} \right) = \left[ M^2L^2T^{-4} \right]

uG=[M1L1T−2]\sqrt{uG} = \left[ M^1L^1T^{-2} \right]

This corresponds to the dimensions of force per unit mass. Therefore, Option (2) is correct.