A mass of 50 kg is placed at the center of a uniform spherical shell of mass 100 kg and radius 50 m. If the gravitational potential at a point, 25 m from the center is $$V$$ kg m$$^{-1}$$. The value of $$V$$ is:

We are asked to find the gravitational potential at a point that lies inside a thin, uniform spherical shell of mass $$100\ \text{kg}$$ and radius $$50\ \text{m}$$, while a point mass of $$50\ \text{kg}$$ sits exactly at the common centre. The point of interest is situated at a distance $$25\ \text{m}$$ from this centre.

We recall two standard results from Newtonian gravitation:

1. Potential due to a point mass. For a point mass $$m$$, the gravitational potential at a distance $$r$$ is given by the formula $$\displaystyle V_{\text{point}} = -\,\frac{G\,m}{r}$$ where $$G$$ is the universal gravitational constant.

2. Potential inside a uniform spherical shell. At any interior point (including the very centre) of a thin spherical shell of total mass $$M$$ and radius $$R$$, the potential has the same constant value $$\displaystyle V_{\text{shell (inside)}} = -\,\frac{G\,M}{R}.$$ It is independent of how deep inside we are, as long as we remain inside the hollow shell.

Because potential is a scalar quantity, the net potential is simply the algebraic sum of the individual potentials contributed by each mass.

We first compute the shell’s contribution. Here $$M = 100\ \text{kg}, \qquad R = 50\ \text{m}.$$ Using the stated formula, we have $$V_{\text{shell}} = -\,\frac{G\,M}{R} = -\,\frac{G \times 100}{50} = -\,\frac{100}{50}\,G = -\,2\,G.$$

Next we compute the point mass’s contribution. For the point mass at the centre, $$m = 50\ \text{kg}, \qquad r = 25\ \text{m}.$$ Using the point-mass formula, $$V_{\text{point}} = -\,\frac{G\,m}{r} = -\,\frac{G \times 50}{25} = -\,\frac{50}{25}\,G = -\,2\,G.$$

Now we add the two potentials:

$$\begin{aligned} V_{\text{total}} & = V_{\text{shell}} + V_{\text{point}} \\ & = (-\,2\,G) + (-\,2\,G) \\ & = -\,4\,G. \end{aligned}$$

Thus the gravitational potential at the given point is

$$V = -\,4\,G.$$

Hence, the correct answer is Option C.