Question

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

• A. Pressure gradient per unit mass
• B. Force per unit mass
• C. Gravitational potential
• D. Energy per unit mass

Answer 1

The Correct Option is B

Dimensional analysis:

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

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

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

Answer 2

Step 1: Write the dimensions of the given quantities
Gravitational constant \( G \) has dimensional formula: \[ [G] = [M^{-1} L^3 T^{-2}]. \] Energy density \( u \) (energy per unit volume) has dimensional formula: \[ [u] = [M L^{-1} T^{-2}]. \]

Step 2: Find the dimensions of \( \sqrt{uG} \)
Multiply the dimensions: \[ [uG] = [M L^{-1} T^{-2}] \times [M^{-1} L^3 T^{-2}] = [L^2 T^{-4}]. \] Now take the square root: \[ [\sqrt{uG}] = [L^1 T^{-2}]. \]

Step 3: Identify the physical quantity with same dimensions
A quantity having dimensional formula \([L T^{-2}]\) represents acceleration or force per unit mass (since \( F/m \) has the same dimensions).

Step 4: Conclusion
Hence, the quantity that has the same dimension as \( \sqrt{uG} \) is force per unit mass (or acceleration).

Final answer

Force per unit mass