If $$G$$ be the gravitational constant and $$u$$ be the energy density then which of the following quantity have the dimensions as that of the $$\sqrt{uG}$$ :

We need to find the dimensions of $$\sqrt{uG}$$ where $$u$$ is energy density and $$G$$ is the gravitational constant.

Dimensions of energy density $$u$$.

$$ [u] = \frac{\text{Energy}}{\text{Volume}} = \frac{[ML^2T^{-2}]}{[L^3]} = [ML^{-1}T^{-2}] $$

Dimensions of gravitational constant $$G$$.

From $$F = \frac{Gm_1m_2}{r^2}$$:

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

Dimensions of $$\sqrt{uG}$$.

$$ [uG] = [ML^{-1}T^{-2}] \times [M^{-1}L^3T^{-2}] = [M^0L^2T^{-4}] = [L^2T^{-4}] $$

$$ [\sqrt{uG}] = [LT^{-2}] $$

Identify the physical quantity.

$$[LT^{-2}]$$ is the dimension of acceleration, which is force per unit mass.

The correct answer is Option (4): Force per unit mass.