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Chapter 9: Problem 34

A uniform ring of mass \(m\) is lying at a distance \(1.73 a\) from the centre of a sphere of mass \(M\) just over the sphere where \(a\) is the small radius of the ring as well as that of the sphere. Then gravitational force exerted is (a) \(\frac{G M m}{8 a^{2}}\) (b) \(\frac{G M m}{(1.73 a)^{2}}\) (c) \(\sqrt{3} \frac{G M m}{a^{2}}\) (d) \(1.73 \frac{G M m}{8 a^{2}}\)

Short Answer

Expert verified
None of the given options are correct as per the application of Newton's law of universal gravitation. The gravitational force, based on the given data, should be \( \frac{G M m}{3a^{2}}\), which is not among the choices.

Step by step solution

01

Understand and visualize the problem

The first step is to understand the problem and visualize it. Draw a diagram of the problem scenario. We have a sphere of mass \(M\) and a ring of mass \(m\), which lies at a distance of \(1.73a\) from the center of the sphere. The small radius of the ring and sphere is \(a\).
02

Apply Newton's law of universal gravitation

According to Newton's law of universal gravitation, the force \(F\) between two bodies is given by \(F = \frac{G M m}{r^{2}}\), where \(G\) is the gravitational constant, \(M\) and \(m\) are the masses of the bodies and \(r\) is the distance between the centers of the two bodies. In this case, \(M = M\), \(m = m\) and \(r = 1.73a\). Substituting these values into the formula gives \(F = \frac{G M m}{(1.73a)^{2}}\).
03

Simplify

Evaluate the force. Simplifying, we get \( F = \frac{G M m}{3a^{2}}\), which isn't an option in the given choices. Therefore, the exercise might contain a mistake.

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