If the distance between earth and the sun were half of its present value, then how many number of days will be there in a year?
Hint; use $T^{2} \propto R^{3}$ i.e., Kepler's third law.

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Solution

Given,

Time period of full rotation around earth, $T_{1} = 365 d a y s$

By Kepler's law

$T^{2} α a^{3}$

$\frac{T_{1}^{2}}{a_{1}^{3}} = \frac{T_{2}^{2}}{a_{2}^{3}}$

$T_{2}^{2} = {(\frac{a_{2}}{a_{1}})}^{3}$ $\times T_{1}^{2}$

$T_{2}^{2} = {(\frac{1}{2})}^{3}$ $\times T_{1}^{2}$

$T_{2}^{2} = (\frac{1}{8})$ $\times 1 y e a r$

$T_{2} = \frac{1}{\sqrt{8}}$ $\times 365 d a y s = 129 d a y s$

Hence, time period is $129 d a y s$

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