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Question
Three uniform spheres each having a mass M and radius a are kept in such a way that each touches the other two. Find the magnitude of the gravitational force on any of the spheres due to the other two.
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Solution
Three spheres are placed with their centres at A, B and C as shown in the figure.
Gravitational force on sphere C due to sphere B is given by
\[\overrightarrow {F}_{CB} = \frac{G m^2}{4 a^2}\cos 60^\circ \hat i + \frac{G m^2}{4 a^2} \cdot \sin 60^\circ \hat j\]
Gravitational force on sphere C due to sphere A is given by \[\overrightarrow {F}_{CA} = - \frac{G m^2}{4 a^2} \cos 60^\circ\hat i + \frac{G m^2}{4 a^2} \cdot \sin 60^\circ\hat j\]
\[\therefore {\overrightarrow F}_{CB} = \overrightarrow {F}_{CB} + \overrightarrow {F}_{CA} \]
\[ = + \frac{2G m^2}{4 a^2}\sin 60^\circ \hat j \]
\[ = + \frac{2G m^2}{4 a^2} \times \frac{\sqrt{3}}{2}\]
i.e., magnitude \[= \frac{\sqrt{3} G m^2}{4 a^2}\] along CO
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