Write dimensions of the given quantities

We need the dimensions of $\sqrt{uG}$ where:

• $u$ = energy density
• $G$ = gravitational constant

Dimension of energy density $u$

Energy has dimension: $[E] = [ML^2T^{-2}]$

Volume has dimension: $[V] = [L^3]$

Therefore energy density: $[u] = \frac{[E]}{[V]} = \frac{ML^2T^{-2}}{L^3} = [ML^{-1}T^{-2}]$

Dimension of gravitational constant $G$

From Newton's law: $F = \frac{Gm_1m_2}{r^2}$

So, $[G] = \frac{[F][r^2]}{[m]^2} = \frac{(MLT^{-2})(L^2)}{M^2} = [M^{-1}L^3T^{-2}]$

Dimension of $uG$

$[uG] = [ML^{-1}T^{-2}]\,[M^{-1}L^3T^{-2}]$ $= [L^2T^{-4}]$

Dimension of $\sqrt{uG}$

$[\sqrt{uG}] = [L^2T^{-4}]^{1/2} = [LT^{-2}]$

So we need the option having dimensions: $[LT^{-2}]$

Check each option

A: Gravitational potential

Gravitational potential = potential energy per unit mass $\left[\frac{E}{m}\right] = \frac{ML^2T^{-2}}{M} = [L^2T^{-2}]$

Not equal to $[LT^{-2}]$.

B: Pressure gradient per unit mass

Pressure: $[P] = [ML^{-1}T^{-2}]$

Pressure gradient: $\left[\frac{P}{L}\right] = [ML^{-2}T^{-2}]$

Per unit mass: $\left[\frac{ML^{-2}T^{-2}}{M}\right] = [L^{-2}T^{-2}]$

Not equal to $[LT^{-2}]$.

C: Energy per unit mass

$\left[\frac{E}{m}\right] = [L^2T^{-2}]$

Not equal to $[LT^{-2}]$.

D: Force per unit mass

$\left[\frac{F}{m}\right] = \frac{MLT^{-2}}{M} = [LT^{-2}]$

This matches.

Conclusion

The quantity having the same dimensions as $\sqrt{uG}$ is: $\boxed{\text{D: Force per unit mass}}$

Comparison with stored answer

Stored correct answer: D

Our derived answer: D

Hence, they agree.