# 4. Let the period of revolution of a planet at a distance R from a star be T. Prove that if it was at a distance of 2R from the star,

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4. Let the period of revolution of a planet at a distance R from a star be T. Prove that if it was at a distance of 2R from the star, its period of revolution will be $$\sqrt8$$ T.

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Thus, when the period of revolution of the planet at a distance R from a star is T, then from kepler's  third law of planetry motion, we have

$$T^2\propto R^3 \space\space\space\space.........(1)$$

now , when the distance of the planet from the star is 2R, then its period of revolution becomes

$$T_1^2\propto(2R)^3$$

$$T_1^2\propto8R^3$$    ........(2)

Dividing equation 2 by 1

$$\tfrac{T_1^2}{T^2}=\tfrac{8R^3}{R^3}$$

$$T_1^2= 8 T^2$$

$$\therefore T_1=\sqrt8T$$

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