Question

Suppose we have made a model of the solar system scaled down with respect to the distances between the planets and sun and the sizes of them. It is scaled down in the ratio n. But the mean density of materials used are the same as the actual materials of the planets and the sun. Consider the orbital approximately as circular. Then the orbital periods of planetary models

• A. Remains same as the original values.
• B. Changes to $n^{2}T$ where T is the original time period.
• C. Change to $\frac{T}{n^2}$ where T is the original time period
• D. Changes to $n^{\frac{3}{2}}T$, where T is the original time period.

Answer 1

The correct option is A Remains same as the original values.

$\frac{GMm}{r^2}=mr{\omega }^2\Rightarrow \omega =\sqrt{\frac{GM}{r^3}}$
$T=\frac{2\pi }{w}=2\pi \sqrt{\frac{r^3}{GM}}=2\pi \sqrt{\frac{r^3}{G.\rho \frac{4}{3}\pi R^3}}=2\pi \sqrt{\frac{3}{4G\rho \pi }}\times {\left(\frac{r}{R}\right)}^{\frac{3}{2}}$
The new time period $=2\pi \sqrt{\frac{3}{4G\rho \pi }}\times {\left(\frac{\frac{r}{n}}{\frac{R}{n}}\right)}^{\frac{3}{2}}$
$\therefore$ time period remains same.