Question

Two satellites are revolving around the earth in circular orbits of the same radii. Mass of one satellite is $$100$$ times that of the other. Then their periods of revolution are in the ratio:
A. $$100:1$$
B. $$1:100$$
C. $$1:1$$
D. $$10:1$$

Answer 1

Hint: The motion of a satellite has the same characteristics as any object in circular motion. We have learned that, for a uniform circular motion the time does not depend on mass. By analyzing the formula for the time period of a satellite, we can find out the solution to this question.
Formula used:
$$v={\displaystyle \frac{2\pi r}{T}}$$

Complete answer:
We know that, for a uniform circular motion Velocity v is given by,
$$v={\displaystyle \frac{2\pi r}{T}}$$
Then,
$$T={\displaystyle \frac{2\pi r}{v}}$$
Where,
$$T$$ is the time taken and $$r$$ is the radius.
It is clear from the formula that time period does not depend on mass.
Hence, the periods of revolution of the satellites are in the ratio $$1:1$$

So, the correct answer is “Option C”.

Additional Information:
A satellite is an object in which only one force is acting. This governing force is the gravitational force. Gravitational force acts as a centripetal force. The motion of a satellite can be described by the same motion characteristics as any object in circular motion. Satellite is an example of projectile. It will continue its motion as long as gravity is present. In the absence of gravitational force it will move in a straight line tangent to earth.

Note:
This can be proved in another way.
The time period of a satellite is given by,
$$T=2\pi \sqrt{{\displaystyle \frac{R^2}{GM}}}$$
Where, $$G$$ is constant, $$R$$ is orbital radius, $$M$$ is mass of earth.
Therefore, the time period of a satellite is independent of its mass.
Hence, two satellites revolving around the earth in circular orbits of the same radii, with one having mass 100 times that of the other, then their periods of revolution are in the ratio $$1:1$$.