Question 83: Gravitation

Question

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 $$\sqrt{\mathrm{uG}}$$ :

Accepted Answer

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 $$\sqrt{\mathrm{uG}}$$.

Answer

Gravitational potential

Answer

pressure gradient per unit mass

Answer

Energy per unit mass

Answer

Force per unit mass

Gravitation

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

Choose the correct answer: