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Question

A planet of mass $$m$$ moves around the sun of mass $$M$$ in an elliptical orbit. The maximum and minimum distance of the planet from the sun are $$r_1$$ and $$r_2$$ respectively. The time period of the planet is proportional to


A
r2/51
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B
(r1+r2)3/2
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C
(r1r2)3/2
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D
r3/2
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Solution

The correct option is A $$\displaystyle \left ( r_1+r_2 \right )^{3/2}$$
Kepler's third law:
Square of the time period $$T$$ is directly proportional to cube of semi-major axis i.e mean distance of the planet $$ \dfrac { { r }_{ 1 }+{ r }_{ 2 } }{ 2 } $$
$${ T }^{ 2 }\propto { \left( \dfrac { { r }_{ 1 }+{ r }_{ 2 } }{ 2 }  \right)  }^{ 3 }$$

$${ T }\propto { \left( { r }_{ 1 }+{ r }_{ 2 } \right)  }^{ 3/2 }$$

Physics

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