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 $$ML^{2}T^{-2}$$ (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:

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

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

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

So, the dimension of $$\sqrt{uG}$$ is $$LT^{-2}$$, 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 $$LT^{-2}$$.
• 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.
• 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.
• 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.

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