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Step-by-Step Solution
1. Understanding the Problem
We have a uniform spherical shell of mass $100\,\text{kg}$ and radius $50\,\text{m}$. A mass of $50\,\text{kg}$ is placed at the center of this shell. We want to find the gravitational potential at a point that is $25\,\text{m}$ from the center. Since we are inside a spherical shell, its contribution to the potential remains uniform throughout the inner region, and we also need to add the contribution due to the centrally placed mass.
2. Potential Contribution from the Central Mass
The gravitational potential at a distance $r$ from a point mass $m$ is given by
$-\frac{Gm}{r}$.
Here, the mass at the center is $50\,\text{kg}$, and the distance is $25\,\text{m}$. Therefore:
$$V_{\text{center}} = -\frac{G \times 50}{25} \;=\; -2G.$$
3. Potential Contribution from the Shell
For a uniform spherical shell of mass $M$ and radius $R$, the gravitational potential at any point inside the shell (including at the center) is the same as that on the surface, given by
$-\frac{GM}{R}$.
Here, the shell has mass $100\,\text{kg}$ and radius $50\,\text{m}$. So its potential contribution is:
$$V_{\text{shell}} = -\frac{G \times 100}{50} \;=\; -2G.$$
4. Summation of Potentials
The total potential at the point $25\,\text{m}$ from the center is the sum of both contributions:
$$V_{\text{total}} = V_{\text{center}} + V_{\text{shell}} = -2G + (-2G) = -4G.$$
5. Conclusion
The gravitational potential at the given point is $-4G$. Hence, the correct answer is Option 4.
