Two identical masses, both having mass m1 are fixed in place at (x,y) = (0,L) an

Question

Two identical masses, both having mass m1 are fixed in place at (x,y) = (0,L) and (0,-L). A third mass m2 is initially at the origin and moving in the +x direction with speed vi.

1. Give the equation that you would use to solve for the maximum distance in the +x direction that m2 will go. Clean it up a little bit, but you do not need to solve it for xf because the result is not pretty

2. Find the distance xf if m1 = 1.99?1030 kg, m2 = 5.97?1024 kg, L = 1.50?1011 meters, and vi = 3.00?104 m/s.

3. Describe what happens to this (symmetric and ideal) system in the long-run.

THANKS!

Explanation / Answer

1) The initial KE of m2 is (1/2)mv^2. At the maximum distance it is 0.
The gravitational PE of m2 is initially -GmM/L wrt each fixed mass. Potentials add, so do potential energies, so the total initial PE is -2GmM/L. At the maximum distance X along the x axis, the PE is -GmM/D, where D is the diagonal distance from one fixed mass to the point (X,0).

Conservation of Energy :
-2GmM/L+ (1/2)mv^2 = -2GmM/D + 0
1/L - v^2/(4GM) = 1/D
This equation enables us to find diagonal distance D. The maximum distance along the x axis is X where
D^2 = X^2+L^2.

2) This is just a matter of putting in the numbers and solving.

3) m2 oscillates between +/-X along the x axis. But the restoring force is not proportional to distance from origin, so the motion is not SHM.

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