Problems for Intermediate Methods in Theoretical Physics Edward F. Redish

Three Coupled Oscillators

Three identical carts are connected to exterior walls by four identical springs as shown in the figure.

(a) Using coordinates, yi that are horizontal, have their positive direction to the right, and have their 0 at the i-th cart's equilibrium position, (i = 1, 2, 3) write the laws of motion for the three carts starting with Newton's second law. Ignore friction and air resistance.

(b) Find the normal modes and natural frequencies for this system. Express your frequencies as a multiple of the natural frequency parameter, ω0 = (k/m)1/2.

(c) Explain the motion of the carts in each normal mode and, if you can, explain physically why the frequencies associated with each mode are what they are what they are.

(d) How would your equations differ if you used the same coordinate system for all the masses? Say, choosing the origin at the left wall with the positive direction running to the right? The equations in these coordinates should look dramatically different from those you found for (a). Are they in fact the same? If they are, show it. If they are not, explain why not.