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

<p>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).</p> <p>The gravitational constant $G$ has dimensions given by: <p>$[G] = M^{-1}L^{3}T^{-2}$</p> <p>where $M$ stands for mass, $L$ for length, and $T$ for time.</p></p> <p>Energy density $u$ is defined as the energy per unit volume. Since energy has dimensions of $ML^{2}T^{-2}$ (from the dimension of work or energy, which is force times distance, and force itself has dimension $MLT^{-2}$), and volume has dimensions of $L^{3}$, the dimensions of energy density would be: <p>$[u] = \frac{ML^{2}T^{-2}}{L^{3}} = M L^{-1} T^{-2}$</p></p> <p>Now, we find the dimensions of $\sqrt{uG}$ by multiplying the dimensions of $u$ and $G$, and then taking the square root:</p> <p>$$[\sqrt{uG}] = \sqrt{[u][G]} = \sqrt{(M L^{-1} T^{-2})(M^{-1}L^{3}T^{-2})} = \sqrt{L^{2}T^{-4}} = LT^{-2}$$</p> <p>So, the dimension of $\sqrt{uG}$ is $LT^{-2}$, which corresponds to acceleration (length per square time).</p> <p>Now, let's match this with the provided options:</p> <ul> <li>Option A (Gravitational potential) has dimensions of $[L^{2}T^{-2}]$, not matching our target of $LT^{-2}$.</li> <li>Option B (Pressure gradient per unit mass) would have dimensions of $[M^{-1}L^{-2}T^{-2}][L^{-1}]$ ($Pressure\ Gradient = \frac{Pressure}{Length} = \frac{ML^{-1}T^{-2}}{L}$, and then divided by mass, $M$), which simplifies to $L^{-3}T^{-2}M^{-1}$, not matching.</li> <li>Option C (Energy per unit mass) has dimensions $ML^{2}T^{-2}M^{-1}$ which simplifies to $L^{2}T^{-2}$, also not a match for the target dimension.</li> <li>Option D (Force per unit mass) has dimensions $MLT^{-2}M^{-1}$ which simplifies directly to $LT^{-2}$, an exact match for our target dimension.</li> </ul> <p>Thus, the correct answer is Option D (Force per unit mass), which has the same dimensions as that of $\sqrt{\mathrm{uG}}$. </p>