Gravitational force on three mutually touching spheres – Rankers Physics

Newton's Law of Gravitation: Practice Problem & Solution

If three uniform spheres, each having mass \(M\) and radius \(R\), are kept in such a way that each touches the other two, the magnitude of the gravitational force on any sphere due to the other two is
\(\frac{GM^2}{4r^2}\)
\(\frac{2GM^2}{r^2}\)
\(\frac{2GM^2}{4r^2}\)
\(\frac{\sqrt{3}GM^2}{4r^2}\)

Short Solution (Best for NEET):

To solve this problem, we apply the core principles of Newton's Law of Gravitation. Understanding the underlying formula is key to arriving at the correct answer below:

The distance between the centers of any two touching spheres is \(2R\). The gravitational force between any two is \(F = \frac{GM^2}{(2R)^2} = \frac{GM^2}{4R^2}\). The angle between the two forces acting on one sphere is \(60^\circ\). Net force is \(F_{\text{net}} = \sqrt{3}F = \frac{\sqrt{3}GM^2}{4R^2}\).

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