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.

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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