Sheldon seemed somber. His super satellite, Segundo, in synchronous syzygy over Sumatra was wobbling. Segundo's mass was 4980 kilograms. The orbit was 40,000 kilometers above Sumatra.

1. What is Segundo's potential energy? ( find g for the position of Segundo in its orbit and average that with g at the Earth's surface.)
2. How much work is done on Segundo by gravity


And Thank you!

1. The potential energy of a body is given by:

V = \frac{Gmm'}{r}

m is the mass of the Earth,
m' is the mass of Segundo,
r is the distance of Segundo from the centre of the Earth,
G is the universal gravitational constant.

2. Work done is the energy lost by Segundo. If gravity did work, that means that gravity has pulled Segundo towards itself. Which in turn means that the lost in potential energy of Segundo when falling is the work done by gravity.

If Segundo didn't fall at all, then there is no loss in potential energy and work done is zero.

Interesting that they suggest you calculate a "local g" for the satellite - by which I suppose they mean g = Gm/r^2, where m = mas of earth and r = radius from center of earth to the satellite. I don't understand why they suggest that. Unk's approach is far easier to calculate and to understand - my only nit pick is that an object's absolute potential energy in a gravitational field actually a negative number, and the important measure is that the change in potential energy as an object moves. The change in potential energy of the satellite as it was launched from the earth to its orbit is:


\Delta PE = \int _{R_1} ^{R_2} \frac {Gmm'} {r^2} dr = -Gmm'(\frac 1 {R_2} - \frac 1 {R_1} )


where R_1 = radius of earth, and R_2 = radius of the satellite's orbit

Side note: the word "syzygy" is misused in this problem. In astronomy "syzygy" is a situation where three objects are aligned in roughly a sraight line - such as occurs during a lunar or solar eclipse. The correct word that should have been used in this problem is "orbit," as in "synchronous orbit over Sumatra."