# Harmonic oscillation, spring not attached to center of mass

1. Oct 21, 2007

### Antti

A bar is guided so that one end moves on a vertical line and the other on a horizontal line. A spring is attached to the upper end according to the figure. Any friction is neglected.

http://web.comhem.se/~u48800174/springbar.jpg [Broken]

I want to find out how x varies in time. If the center of mass would have been at x and k is the spring coefficient and m the mass, then

$$x = A sin (\sqrt{k/m}t + \phi_{0})$$

But now the center of mass is at the center of the rod (with length d) instead. More specifically:

$$\overline{CM} = (x/2, -\sqrt{d^2 - x^2}/2$$

How does this affect the equation of motion? Is it perhaps still the same? My intuition tells me something would change when the spring isn't attached at the COM but i don't know what and why. Thankful for help!

Last edited by a moderator: May 3, 2017
2. Oct 21, 2007

### Meir Achuz

This problem is easiest with Lagrange's equations.
If you don't know them, write a good free body diagram.