A uniform metal sphere of radius a and mass M is surrounded by a thin uniform spherical shell of equal mass and radius 4a (In the following figure). The centre of the shell falls on the surface of the inner sphere. Find the gravitational field at the points P_{1} and P_{2 }shown in the figure.

#### Solution

At point P_{1}, the gravitational field due to the sphere and the shell is given by*F \[= \frac{GM}{\left( 3a + a \right)^2} + 0 = \frac{GM}{16 a^2}\] *

At point P_{2}, the gravitational field due to the sphere and the shell is given by

\[F = \frac{GM}{\left( a + 4a + a \right)^2} + \frac{GM}{\left( 4a + a \right)^2}\]

\[ \Rightarrow F = \frac{GM}{36 a^2} + \frac{GM}{25 a^2}\]

\[ \Rightarrow F = \frac{GM}{a^2}\left( \frac{1}{36} + \frac{1}{25} \right)\]

\[ \Rightarrow F = \frac{GM}{a^2}\left( \frac{25 + 36}{900} \right)\]

\[ \Rightarrow F = \left( \frac{61}{900} \right)\frac{GM}{a^2}\]