SHM

[tukeywilliams]

A block is hung on a spring, and the frequency f of the oscillation of the system is measured. The block, a second identical block, and the spring are carried into space. The 2 blocks are attached to the ends of a spring, and the system is taken out into space on a space walk. The spring is extended, and the system is released to oscillate while floating in space. What is the frequency of oscillation for this system, in terms if f
I know the answer is sqrt(2)f . But how do we get this?

Thanks

[turin]

Go back to the equation(s) of motion. There will be a two coupled differential equations for the two masses. You can do a clever substitution/change of variables to decouple the equations into a simple harmonic oscillator and a free particle equation (in other words, you diagonalize the differential operator).

Recall the equation of motion for the single simple harmonic oscillator is:

d^2y/dt^2 = -(k/m)(y - y_0) -> (-&omega^2)y = -(k/m)(y - y_0).

The first step to realizing how to generalize this to coupled oscillators is not to ignore the y_0, even though it seems unimportant in the single oscillator case. Basically, in the coupled oscillator case, y_0 can also change, and so the difference becomes important. You can think of the two equations as a single 2x2 matrix equation, and then solving for the frequency is just an eigenvalue problem.

..............[...y....].........[..(-k/m)....(+k/m)..]....[...y....]
(d/dt)^2..[.........]...=...[.............................]...[.........]
..............[..y_0..].........[..(+k/m)....(-k/m)..]...[..y_0..]

Solve for the diagonal modes.

Sorry, the post is not displaying my spaces faithfully, so I had to stuff it with a bunch of periods to keep it from deflating. It still doesn't look quite right.

This post has been edited by turin on Mar 20 2007, 05:27 AM