# An antenna loosens and becomes detached from a satellite in a circular orbit around the earth....

## Question:

An antenna loosens and becomes detached from a satellite in a circular orbit around the earth. Describe the antenna's subsequent motion. If it will land on earth, describe where: if not, describe how it would be made to land on earth.

## Orbital Motion of a Satellite

A satellite in the orbit is revolving the planet at some definite orbital velocity and this orbital velocity is determined by equating the gravitational force acting on the satellite and centripetal force of the satellite due to its circular motion about the planet. The orbital velocity can be expressed as {eq}v = \sqrt { \dfrac { G M } { r } } {/eq}. Here G, M and r are the mass of the planet, gravitational constant, and radius of the orbit of the satellite respectively. When the satellite is launched into an orbit, the satellite is given sufficient orbital velocity as calculated above. The satellite will continue its orbital motion as the angular momentum of the satellite is conserved.

Given data

• Antenna looses from the satellite and gets detached.

The antenna was also in orbital motion with same orbital speed in the orbit. So when it gets detached it will have sufficient orbital speed to remain in the same orbit.

We can directly see that orbital speed does not depend on the mass of the object. Orbital speed {eq}v = \sqrt { \dfrac { G M } { r } } {/eq}

Here M and r are the mass of the planet and radius of the orbit.

The constant G is the gravitational constant.

So the detached antenna also will orbit the earth in the same orbit just like the satellite and it will not fall back to earth.

In order to make the antenna to fall on to earth, we need to reduce its orbital speed. Once the orbital speed is less, the gravitational force acting on the antenna will become greater than the centripetal force and the antenna will spiral down to the earth. 