Energy in a system of orbiting satellite
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Energy in a system of orbiting satellite

[From: ] [author: ] [Date: 11-12-02] [Hit: ]
B.What is the change in the systems kinetic energy?C.What is the change in the systems potential energy?-A. There are no nonconservative forces acting on the satellite,......
A 1 048-kg satellite orbits the Earth at a constant altitude of 110-km.
A.How much energy must be added to the system to move the satellite into a circular orbit with altitude 210 km?
B.What is the change in the system's kinetic energy?
C.What is the change in the system's potential energy?

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A. There are no nonconservative forces acting on the satellite, so the mechanical energy is conserved going from orbit 1 into orbit 2 except for the added energy. Note that E1 = PE1 + KE1 and E2 = PE2 + KE2 where E is the mechanical energy, PE is the gravitational potential energy, and KE is the kinetic energy.

We have that PE = mg(R + h) where R is the radius of the Earth (= 6.38 x 10^6 m), h is the height of the orbit and m is the mass of the satellite. We can assume g does not change by going from orbit 1 to orbit 2 since the height difference is very small. Also we have KE = 1/2 m v^2 ignoring any rotational effects. It is given that h1 = 110 km and h2 = 210 km. But what are v1 and v2?

The velocity v can be calculated using Newton's Law of Universal Gravitation. Since F = ma by the 2nd law of dynamics, then F = G m M / (R + h)^2 = m v^2 / (R + h) = m a, where G is a universal constant, m is satellite's mass, M is the Earth's mass, and v^2 / (R + h) is the centripetal acceleration for the circular orbit.

If you do a little algebra, you can find that v = sqrt(GM/(R + h)). So to put it all together, we have E1 + added energy = E2 which gives mg(R + h1) + 1/2 m GM/(R + h1) + added energy = mg(R + h2) + 1/2 m GM/(R + h2). Put in the values and calculate the added energy.

B. This is simply 1/2 m (v2)^2 - 1/2 m (v1)^2

C. This is simply mg(R + h2) - mg(R + h1)

I will leave the numerical calculations to yourself.

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I'm so dumbfounded, I don't even know what to say. If you figure it out, I'm going to start praying to you.
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