A uniform metal sphere of radius $a$ and mass $M$ is surrounded by a thin uniform spherical shell of

Question

A uniform metal sphere of radius $a$ and mass $M$ is surrounded by a thin uniform spherical shell of equal mass and radius $4a$. The centre of the shell falls on the surface of the inner sphere. What are the magnitudes of the gravitational fields at points $P_1$ and $P_2$, which lie on the line passing through the centres at distances $4a$ and $6a$ respectively from the solid sphere's centre? (Assume $P_1$ is located inside the shell and $P_2$ is located outside the shell)

Professor PC Thomas Classes · Answer

Let the solid sphere's centre be $C_1$ and the shell's centre be $C_2$. Distance $C_1C_2 = a$. Point $P_1$ is at $4a$ from $C_1$ and $3a$ from $C_2$. Since $3a < 4a$, $P_1$ is inside the shell, so the shell exerts zero field. Field at $P_1 = \frac{GM}{(4a)^2} = \frac{GM}{16a^2}$. Point $P_2$ is at $6a$ from $C_1$ and $5a$ from $C_2$. Since $5a > 4a$, $P_2$ is outside the shell. Field at $P_2 = \frac{GM}{(6a)^2} + \frac{GM}{(5a)^2} = \frac{GM}{36a^2} + \frac{GM}{25a^2} = \frac{61GM}{900a^2}$.