Question

Two planets $\text{A}$ and $\text{B}$ travel counterclockwise in circular orbits around a fixed star. The radii of their orbits are in the ratio $1:4$. At some time, they are aligned as shown in the figure, making a straight line with the star. After a certain time, the planet $\text{A}$ comes back to its initial position, completing one full circle about the star. In the same time, angular displacement of the planet $\text{B}$ is

• A. ${22.5}^{\circ }$
• B. ${45}^{\circ }$
• C. ${180}^{\circ }$
• D. ${360}^{\circ }$

Answer 1

The correct option is B ${45}^{\circ }$

Given:
$r_A:r_B=1:4$

As per Kepler’s law of periods,

$T^2\propto R^3$

So, $\frac{T_A}{T_B}={\left(\frac{r_A}{r_B}\right)}^{3/2}=\frac{1}{8}$

As angular displacement of planet $\text{A}$ is ${360}^{\circ }$, angular displacement of $\text{B}$ will be

$\theta =\frac{{360}^{\circ }}{8}={45}^{\circ }$

Hence, option $\left(b\right)$ is correct.