Question

A geostationary satellite is moving in a circular orbit at a height 36000km from the surface of earth. calculate its speed if it takes 24 hours to revolve around the earth?
(give radius of earth equal 6400km )

Answer 1

When satellites are revolving around bigger planets those are acted upon gravitational force due to attraction between planet and satellite. This gravitational force of attraction is the reason for the revolution as this provides the required centripetal force. By equating to centripetal force we can get velocity and then time period or else if we already know the time period and radius then we can find the velocity.

Complete step by step solution:
For a satellite there are two ways to find out the time period and velocity. First we know the mass of the planet around which it is rotating and radius of rotation of the satellite and we know the gravitational constant too. Then we will equate the centripetal force to the gravitational force of attraction between satellite and the earth and we will find out the velocity of rotation. After finding out the velocity we have radius of rotation so we can find out time period from the below equation
Time period of revolution(T) = $2\pi R/v$
For the given question all of the above procedures are not required. We have a radius of rotation of the satellite and time period of rotation. Hence we can directly get velocity of rotation too.
Radius of rotation is $36000 + 6400 = 42400Km$ and the time period of rotation is 24 hours.
Time period will be $2\pi R/v$
\eqalign{ & T = 2\pi R/v \cr & \Rightarrow v = \dfrac{{2\pi R}}{T} \cr & \Rightarrow v = \dfrac{{2\pi \left( {42400} \right)}}{{24}} \cr & \Rightarrow v = \dfrac{{2\left( {3.14} \right)\left( {42400} \right)}}{{24}} \cr & \Rightarrow v = \dfrac{{266272km}}{{24h}} \cr & \therefore v = 11094.67km/h \cr & \cr}
Hence the velocity of revolution will be 11094 kilometer per hour.

Note: Geo stationary satellite in the sense if we see the satellite from the earth it appears to be at rest. This is because the angular velocity of revolution of satellite is the same as angular velocity of rotation of earth. Hence time periods of earth and satellite will be the same i.e 24 hours but the linear velocities will be different as the radius of rotations is different.