Gravitational force and laws

Question

If the distance between the earth and the sun becomes half its present value, the number of days in a year would have been

Moderate

Solution

According to Kepler's third law, the ratio of the squares of the periods of any two planets revolving about the sun is equal to the ratio of the cubes of their average distances from the sun i.e.

$\begin{array}{l}{\left(\frac{{\mathrm{T}}_{1}}{{\mathrm{T}}_{2}}\right)}^{2}={\left(\frac{{\mathrm{r}}_{1}}{{\mathrm{r}}_{2}}\right)}^{3}={\left[\frac{{\mathrm{r}}_{1}}{\frac{1}{2}{\mathrm{r}}_{1}}\right]}^{3}=8\Rightarrow \frac{{\mathrm{T}}_{1}}{{\mathrm{T}}_{2}}=2\sqrt{2}\\ \therefore {\mathrm{T}}_{2}=\frac{{\mathrm{T}}_{1}}{2\sqrt{2}}=\frac{365\text{days}}{2\sqrt{2}}=129\text{days}\end{array}$

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