Question

If the distance between the Earth and the Sun was half of its present value, what will be the approximate number of days in a year on the Earth?

Answer 1

Let the present time period of revolution of the earth be $T_1$, i.e. $365days$
Let $T_2$ be the time period of revolution when the distance between the Sun and the Earth is halved.
Let $R_1$ be the present radius of revolution and $R_2$ be the radius of revolution when the distance is halved.
$\therefore R_1=2R_2$
According to Kepler's law of period, $R^3\propto T^2⟹T\propto R^{\frac{3}{2}}$

$\therefore \frac{T_1}{T_2}={\left(\frac{R_1}{R_2}\right)}^{\frac{3}{2}}={\left(\frac{R_1}{\frac{R_1}{2}}\right)}^{\frac{3}{2}}=2\sqrt{2}$
$\Rightarrow T_2=\frac{T_1}{2\sqrt{2}}=\frac{365}{2\sqrt{2}}days=129days$

Hence, the time period of revolution of the earth will be $129days$ if the distance between the Earth and the Sun is halved.