The total mechanical energy of a satellite of mass m orbiting the earth at a distance equal to 2 times the earth's radius above the surface is E = - GMm/4R
To find what is the total mechanical energy of a satellite of mass m orbiting the earth at a distance equal to 2 times the earth's radius above the surface, we proceed as follows
We know that the total mechanical energy of the satellite is E = K + U where K = kinetic energy = 1/2mv² where
- m = mass of satellite and
- v = speed of satellite
and U = gravitational potential energy = -GMm/r
where
- G = universal gravitational constant
- M = mass of earth
- m = mass of satellite and
- r = radius of orbit
So, E = K + U
E = 1/2mv² + (-GMm/r)
E = 1/2mv² - GMm/r
Now, we know that the gravitational force on the satellite due to the earth is F = GMm/r². This force is equal to the4 centripetal force at that point F' = mv²/r.
So, F = F'
GMm/r² = mv²/r
GMm/r = mv²
Dividing both sides by 2, we have that
GMm/2r = mv²/2
Substituting this into the equation for E, we have that
E = 1/2mv² - GMm/r
E = GMm/2r - GMm/r
E = (GMm - 2GMm)/2r
E = - GMm/2r
Now, given that
- r = 2R where R = radius of earth
E = - GMm/2r
E = - GMm/2(2R)
E = - GMm/4R
So, the total mechanical energy is E = - GMm/4R