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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 (In the following figure). The centre of the shell falls on the surface of the inner sphere. Find the gravitational field at the points P1 and P2 shown in the figure.
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Solution
At point P1, 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 P2, the gravitational field due to the sphere and the shell is given by
\[F = \frac{GM}{\left( a + 4a + a \right)^2} + \frac{GM}{\left( 4a + a \right)^2}\]
\[ \Rightarrow F = \frac{GM}{36 a^2} + \frac{GM}{25 a^2}\]
\[ \Rightarrow F = \frac{GM}{a^2}\left( \frac{1}{36} + \frac{1}{25} \right)\]
\[ \Rightarrow F = \frac{GM}{a^2}\left( \frac{25 + 36}{900} \right)\]
\[ \Rightarrow F = \left( \frac{61}{900} \right)\frac{GM}{a^2}\]
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