Solution Found!
Consider two particles interacting by a Hooke's law
Chapter 8, Problem 8.11(choose chapter or problem)
Consider two particles interacting by a Hooke's law potential energy, U = Ikr2, where r is their relative position r = r1 r2, and subject to no external forces. Show that r(t) describes an ellipse. Hence show that both particles move on similar ellipses around their common CM. [This is surprisingly awkward. Perhaps the simplest procedure is to choose the xy plane as the plane of the orbit and then solve the equation of motion (8.15) for x and y. Your solution will have the form x = A cos cot B sin cot, with a similar expression for y. If you solve these for sin cot and cos cot and remember that sine cos2 = 1, you can put the orbital equation in the form axe 2bxy cy2 = k where k is a positive constant. Now invoke the standard result that if a and c are positive and ac > b2, this equation defines an ellipse.]
Questions & Answers
QUESTION:
Consider two particles interacting by a Hooke's law potential energy, U = Ikr2, where r is their relative position r = r1 r2, and subject to no external forces. Show that r(t) describes an ellipse. Hence show that both particles move on similar ellipses around their common CM. [This is surprisingly awkward. Perhaps the simplest procedure is to choose the xy plane as the plane of the orbit and then solve the equation of motion (8.15) for x and y. Your solution will have the form x = A cos cot B sin cot, with a similar expression for y. If you solve these for sin cot and cos cot and remember that sine cos2 = 1, you can put the orbital equation in the form axe 2bxy cy2 = k where k is a positive constant. Now invoke the standard result that if a and c are positive and ac > b2, this equation defines an ellipse.]
ANSWER:
Step 1 of 6
The potential energy of a spring with spring constant
In Cartesian coordinate,
So, the potential energy is,
