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Question

A planet of small mass m moves around the Sun of mass M along an elliptical orbit such that its minimum and maximum distances from sun are r and R respectively. Its period of revolution will be:

A
2π(r+R)36GM
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B
2π(r+R)33GM
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C
2π(r+R)38GM
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D
2π(r+R)3GM
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Solution

The correct option is C 2π(r+R)38GM
Given,
Planet's mass=m
Mass of Sun=M


We know that if r and R are the shortest and longest distances respectively then, length of semi-major axis a=r+R2From Kepler's law, T=2πa3GM

Substituting the value of a in the expression for time period.

T=2π(r+R2)3×1GM

T=2π(r+R)38GM

Hence, option (c) is the correct answer.
The time period of revolution of a planet in elliptical orbit is given by T=2πa3GMWhere a is the length of the semi-major axis as well as the mean distance from the sun.

Note: from properties of ellipse

r=a(1e)...(1) and R=a(1+e)...(2)

From equation (1) and (2),a=r+R2

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