Question

Two planets are revolving around a star in circular orbits. If the ratio of radii of orbit is $1:4$, then ratio of their time period will be

• A. $1:1$
• B. $1:4$
• C. $1:8$
• D. $1:16$

Answer 1

The correct option is C $1:8$

$\frac{GM{m}_1}{{r_1}^2}=\frac{m{v_1}^2}{r_1}$

$\Rightarrow {v_1}^2=\frac{GM}{r_1}$

$\Rightarrow v_1=\sqrt{\frac{GM}{r_1}}$ and $\Rightarrow v_2=\sqrt{\frac{GM}{r_2}}$

$\Rightarrow \frac{r_1}{r_2}=\sqrt{\frac{r_1}{r_2}}=2$

$\therefore \frac{t_1}{t_2}=\frac{2\pi r_1}{v_1}\times \frac{v_2}{2\pi r_2}$

$=\left(\frac{r_1}{r_2}\right).\left(\frac{v_2}{v_1}\right)=\frac{1}{8}$

$\Rightarrow \frac{t_1}{t_2}=\frac{1}{8}$