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Phy 3221
Due: March 23, 2011
Homework set # 9
For all of the problems in this set, unless otherwise stated, assume that you have the following
situation: A single object of mass m is attached to the ends of two identical very long springs
of spring-constant k. One spring is lined up on the x-axis, the other on the y-axis. Choose
your axes and positions of the springs so that the equilibrium position of the object is at
x = y = 0. The springs are long enough that if the mass is at xı̂ + y̂ then the restoring
p force
is −k(xı̂+y̂). Assume that there is no damping in this problem. Feel free to let ω = k/m.
Problem 1: Assume that the oscillations in the x and the y directions have the same
amplitude A and are in phase. Describe the object’s path in terms of circular coordinates.
Problem 2: Assume that the oscillations in the x and the y directions have the same
amplitude A but that the oscillations are out of phase by π/2. Describe the object’s path
in terms of circular coordinates.
Problem 3: Assume that the oscillations in the x and the y directions are out of phase
by π/2 and have a small difference in amplitude. To be specific let the amplitude along the
x-axis be A + δA, and along the y axis be A − δA; assume that δA is very small and positive.
a. Describe the object’s path in terms of circular coordinates.
b. What is the frequency of the small radial oscillations of the path?
Problem 4: Assume that the oscillations in the x and the y directions are out of phase by
π/2 and have amplitudes Ax along the x-axis and Ay along the y-axis. Describe the object’s
path in terms of a conic section using either circular coordinates or Cartesian coordinates.
Problem 5: One of Kepler’s laws is that the rate that area (between the sun and the planet)
is swept out by a planet as it orbits the sun is constant, but the rate differs for different
planets. For the problem of an object attached to two springs with different amplitudes, Ax
and Ay , there is a similar law. State this law carefully describing just what area is being
swept out at a constant rate, and find the rate at which the area is being swept out as a
function of k, m, Ax and Ay .