Given: The mass of the earth is 5.97e+24 kg and the radius of the earth is 6.3700e+06 m

Question

Given: The mass of the earth is 5.97e+24 kg and the radius of the earth is 6.3700e+06 m. The gravitational force on an orbiting satellite is the centripetal force.

A. At what height above the ground must a geosynchronous satellite orbit above the equator? A geosynchronous satellite remains over the same point on the surface as the earth rotates and the satellite orbits. (This part is a circular motion problem.)

h =  m

h =  miles

B. If the mass of the satellite is 800 kg, what is the total mechanical energy of this geosynchronous satellite?

E =  J

I don't know how to do this, please help!

Answer 1

Easiest way is to plug 2πR/T for v in the centripetal force equation: F_C = mv²/R = 4π²Rm/T²

Equate the centripetal and gravitational force:

GMm/R² = 4π²Rm/T²

Rearrange and cancel m, the mass of the satellite

T² = (4π²/GM) R³ (which is how Kepler's Law connects to gravitation

Actually you are solving for R given T = 1day (you have to put T in seconds) R will be in meters

You can convert that to get miles.

In the second part, the total mechanical energy is E_K + U_G . You could just plug in 1/2 mv² = E_K and

-GMm/R for U_G

but because mv²/R = GMm/R² E_K = 1/2 |U_G| It is always true that E_Tot = 1/2 U_G So, you only have to plug into the U_G equation.

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