JEE Main 2024 — Gravitation Question with Solution

To determine the dimension of the quantity $$\sqrt{uG}$$, we first need to understand the dimensions of both the gravitational constant (G) and the energy density (u).

The gravitational constant $$G$$ has dimensions given by:

$$[G]=M^{-1}{L}^3{T}^{-2}$$

where $$M$$ stands for mass, $$L$$ for length, and $$T$$ for time.

Energy density $$u$$ is defined as the energy per unit volume. Since energy has dimensions of $$M{L}^2{T}^{-2}$$ (from the dimension of work or energy, which is force times distance, and force itself has dimension $$ML{T}^{-2}$$), and volume has dimensions of $$L^3$$, the dimensions of energy density would be:

$$[u]=\frac{M{L}^2{T}^{-2}}{L^3}=M{L}^{-1}{T}^{-2}$$

Now, we find the dimensions of $$\sqrt{uG}$$ by multiplying the dimensions of $$u$$ and $$G$$, and then taking the square root:

$$[\sqrt{uG}]=\sqrt{[u][G]}=\sqrt{(M{L}^{-1}{T}^{-2})(M^{-1}{L}^3{T}^{-2})}=\sqrt{L^2{T}^{-4}}=L{T}^{-2}$$

So, the dimension of $$\sqrt{uG}$$ is $$L{T}^{-2}$$, which corresponds to acceleration (length per square time).

Now, let's match this with the provided options:

• Option A (Gravitational potential) has dimensions of $$[L^2{T}^{-2}]$$, not matching our target of $$L{T}^{-2}$$.
• Option B (Pressure gradient per unit mass) would have dimensions of $$[M^{-1}{L}^{-2}{T}^{-2}][L^{-1}]$$ ($$PressureGradient=\frac{Pressure}{Length}=\frac{M{L}^{-1}{T}^{-2}}{L}$$, and then divided by mass, $$M$$), which simplifies to $$L^{-3}{T}^{-2}{M}^{-1}$$, not matching.
• Option C (Energy per unit mass) has dimensions $$M{L}^2{T}^{-2}{M}^{-1}$$ which simplifies to $$L^2{T}^{-2}$$, also not a match for the target dimension.
• Option D (Force per unit mass) has dimensions $$ML{T}^{-2}{M}^{-1}$$ which simplifies directly to $$L{T}^{-2}$$, an exact match for our target dimension.

Thus, the correct answer is Option D (Force per unit mass), which has the same dimensions as that of $$\sqrt{\mathrm{uG}}$$.