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Question

A planet of mass m moves in an elliptical path around the Sun so that its maximum and minimum distance from the sun are equal to r1 and r2 respectively. Find the angular momentum of this planet relative to the center of the sun. Mass of the sun is M.

A
mGMr1r2(r1+r2)
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B
m2GMr1r2(r1+r2)
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C
m2GMr21r22(r1+r2)2
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D
mGMr1r22(r1+r2)
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Solution

The correct option is B m2GMr1r2(r1+r2)

Let,
m be the mass of the planet,
v1 be the speed of planet at the farthest point,
v2 be the speed of planet at the closest point.

Conserving angular momentum of m about the Sun.

mv1r1=mv2r2

v2=(r1r2)v1...(1)

Conserving mechanical energy between the closest and the farthest point.

P.E1+K.E1=P.E2+K.E2

GMmr1+12mv21=GMmr2+12mv22

12[v21v22]=GM[1r11r2]

As v2=(r1r2)v1,

[v21v21×r21r22]=GM[1r11r2]

v21[r22r21]r22=2GM[r2r1r1r2]

v21[(r2+r1)(r2r1)r22]=2GM(r2r1)r1r2

v21=2GMr2(r1+r2)r1

v1=2GMr2(r1+r2)r1

Angular momentum

L=mv1r1

L=m×2GMr2(r1+r2)r1×r1

L=m2GMr1r2(r1+r2)

Hence, option (b) is the correct answer.
Why this question?
To make students understand the application of conservation of angular momentum and the conservation of mechanical energy in the same problem. This method is used across various types of problems and hence important to understand well.

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