Question

Two satellites are orbiting around the earth in circular orbits of the same radii. One of them is $10\mathrm{times}$ greater in mass than the other. Their period of revolutions are in the ratio:

• A. 1:100
• B. 100:1
• C. 1:10
• D. 1:1

Answer 1

The correct option is D

1:1

Step 1: The motion of satellite and uniform circular motion

The motion of a satellite can be considered similar to a uniform circular motion.

The velocity of a body rotating in a uniform circular motion can be expressed as follows:

$\mathrm{v}=\frac{2\mathrm{\pi r}}{\mathrm{T}}$

Here,

1. $\mathrm{v}$ is the velocity.
2. $\mathrm{r}$ is the radius of the orbit.
3. $\mathrm{T}$ is the period of revolution.

Step 2: Time period of revolution

Rearranging the equation the period of revolution can be written as follows:

$\mathrm{T}=\frac{2\mathrm{\pi r}}{\mathrm{v}}$

Hence, the revolution period does not depend upon the mass.

Step 3: Ratio of time periods

The radius of the orbits for both satellites is the same.

The ratio of time periods can be determined as follows:

$$\begin{gathered} \frac{{\mathrm{T}}_1}{{\mathrm{T}}_2}=\frac{\frac{2\mathrm{\pi r}}{\mathrm{v}}}{\frac{2\mathrm{\pi r}}{\mathrm{v}}} \\ \frac{{\mathrm{T}}_1}{{\mathrm{T}}_2}=\frac{1}{1} \end{gathered}$$

Therefore, the ratio of the period of revolution will be $1:1$ that is option (d) is correct.