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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.
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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