# 14) A satellite is orbiting the earth. If a payload of material is added until it doubles the...

## Question:

14) A satellite is orbiting the earth. If a payload of material is added until it doubles the satellite?s mass, the earth?s pull of gravity on this satellite will double, but what is the effect on the satellite?s orbit?

A. Not enough information is given to determine the answer.

B. The satellite's orbit will decrease in size by half.

C. The satellite?s orbit will not be affected.

D. The satellite?s orbit will increase in size by half.

## Orbital Motion of a Satellite in a Circular Orbit

A satellite of mass m needs centripetal force to move in a circular orbit of radius r with an orbital speed v around a planet of mass M. The necessary centripetal force {eq}F_c = \dfrac { m v^2 } { r } {/eq} is derived from the gravitational force {eq}F_g = \dfrac { G M m } { r^2 } {/eq} between the satellite and the planet existing between the centers of the two. In this equation G is the gravitational constant. On equating the centripetal force and gravitational force one can get the orbital speed of the satellite or orbital radius of the satellite.

## Answer and Explanation:

Given data

• Initial mass of the satellite m
• Payload in the satellite is doubled and the new mass of the satellite is 2m

Let M and G be the mass of earth and gravitational constant respectively.

Let r be the initial radius of the satellite with orbital speed v when having the mass m

Then the centripetal force acting on the satellite {eq}F_c = \dfrac { m v^2 } { r } {/eq}

Gravitational force acting on the satellite {eq}F_g = \dfrac { G M m } { r^2 } {/eq}

On equating these two we get {eq}\dfrac { m v^2 } { r } = \dfrac { G M m } { r^2 } {/eq}

Therefore radius of the orbit of the satellite {eq}r = \dfrac { G M } { v^2 } {/eq}

Directly we can see that the radius of the orbit is independent of the mass of the satellite.

Based on these we can conclude that the correct option is option c)