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Question

A thin uniform annular disc of mass M has an outer radius 4R and an inner radius 3R as shown in the figure. The work required to take a unit mass from point P on its axis to infinity is



A
2GM7R(425)
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B
2GMR(425)
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C
GM4R
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D
4GM7R(425)
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Solution

The correct option is A 2GM7R(425)
We know that the work required to take a unit mass from P to infinity is VP. i.e., W=VP ​ Where, VP is the gravitational potential at P due to the disc.

To calculate VP let us divide the disc into small elements, each of thickness dr as shown in the figure. Consider one such element at a distance r from the center of the disc.


Mass of the element,

dm=M(2πrdr)π(4R)2π(3R)2=2Mrdr7R2

So, the potential at point P is,

VP=4R3RGdm(4R)2+r2

Substituting value of dm,

VP=2GM7R24R3Rrdr(16R2+r2

Put r2+16R2=x2, we get 2rdr=2xdx rdr=xdx

When r=3R; x2=9R2+16R2; x=5R

When r=4R; x2=16R2+16R2; x=42R

VP=2GM7R242R5Rxdxx2

VP=2GM7R242R5Rdx

VP=2GM7R(425)

From the equation (1),

W=2GM7R(425)

Hence, option (a) is the correct answer.
Why this question: To make the students understand the application of integration to calculete the potential of continuous mass distribution.

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