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 √uG :

Solution

Physics sir · 3 steps

Find the dimensions of uG first, take the square root, and you will recognise an acceleration.

1. Energy density. Energy density is energy per unit volume, so [u] = ML²T⁻²/L³ = ML⁻¹T⁻².
2. Gravitational constant. From F = Gm₁m₂/r², G = Fr²/m₁m₂, so [G] = (MLT⁻² × L²)/M² = M⁻¹L³T⁻².

   [√uG] = √(ML⁻¹T⁻² × M⁻¹L³T⁻²) = √L²T⁻⁴ = LT⁻²

   Option	Quantity	Dimensions
   (a)	Gravitational potential (energy per unit mass)	L²T⁻²
   (b)	Pressure gradient per unit mass	ML⁻¹T⁻²/(L × M) = L⁻²T⁻²
   (c)	Energy per unit mass	L²T⁻²
   (d)	Force per unit mass	MLT⁻²/M = LT⁻²

3. So. √uG has the dimensions of acceleration, LT⁻², which is the same as force per unit mass. The answer is (d).

The idea to keep. Write each quantity in M, L, T first and only then combine them. Energy density is the same as pressure, ML⁻¹T⁻², and force per unit mass is just acceleration.

Answer from the NTA final answer key. Solution by PrepWiser.