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Chapter 6: Problem 751

A particle of mass \(\mathrm{M}\) is situated at the center of a spherical shell of same mass and radius a the magnitude of gravitational potential at a point situated at (a/2) distance from the center will be (A) \([(4 \mathrm{GM}) / \mathrm{a}]\) (B) \((\mathrm{GM} / \mathrm{a})\) (C) \([(2 \mathrm{GM}) / \mathrm{a}]\) (D) \([(3 \mathrm{GM}) / \mathrm{a}]\)

Short Answer

Expert verified
The gravitational potential at a point situated at (a/2) distance from the center is \(-\frac{3GM}{a}\), which corresponds to option (D).

Step by step solution

01

Calculate the gravitational potential due to the central particle

First, let's calculate the gravitational potential due to the mass M placed at the center of the sphere. The formula for gravitational potential V due to a point mass is given by: \[V_p = -\frac{GM}{r}\] Where G is the universal gravitational constant, M is the mass of the central particle, and r is the distance to the point where we are measuring the potential (which is a/2 in this problem). Plugging in the values, we get: \[V_p = -\frac{GM}{(a/2)} = -\frac{2GM}{a}\]
02

Calculate the gravitational potential due to the spherical shell

Next, we will calculate the gravitational potential V due to the spherical shell of mass M and radius a. Inside a uniformly distributed spherical shell, the gravitational potential is constant and equal to the potential at the surface of the shell. To find this potential, we can use the same formula as we used for the central particle mass, but with the radius a instead of a/2: \[V_s = -\frac{GM}{a}\]
03

Add the gravitational potentials

Now we can find the total gravitational potential at the point located at a/2 distance from the center. To do this, we simply add the gravitational potentials due to the central particle and the spherical shell: \[V_{total} = V_p + V_s\] Substituting the values we found in Steps 1 and 2, we have: \[V_{total} = -\frac{2GM}{a} + -\frac{GM}{a} = -\frac{3GM}{a}\]
04

Choose the correct answer

Comparing our result with the given options, we find that the correct answer is: (D) \([(3 \mathrm{GM}) / \mathrm{a}]\)

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