CLASSICAL MECHANICS

DR. KUEHN, WISCONSIN LUTHERAN COLLEGE

Homework 12

Written solutions due Friday, Dec. 4, 2020.

(1) Satellite orbit: A satellite of mass m orbits earth in a circle or radius r0. Oneof its engines is fired briefly toward the center of the earth, changing the energy ofthe satellite but not its angular momentum.

The initial energy of the orbit was Ei, shown in the diagram. The final energy isEf .(a) Write down a formula for the effective potential of the orbit. Don’t make this

complicated.(b) What is the angular momentum, l, of the satellite? Hint: the slope of the

effective potential at r0 is zero.(c) Let’s assume that in the vicinity of r0, the effective potential is approximately

parabolic. This implies that the radial motion (toward and away from the

planet) is simple harmonic motion with frequency ω0 =√k/m, where the

effective spring constant is given by the convexity of the parabola near its

minimum. That is: k =d2Ueff

dr2evaluated at r0. Find ω0. Your answer should

be in terms of l, m, and r0.(d) After firing the rocket, the radial position of the satellite is given by r(t) =

r0 + A sinω0t. Can you qualitatively describe the orbit of this satellite? Howwould the orbit be different if the thruster was in the opposite direction?

1