How Does a Spring Attached to a Pivoting Rod Demonstrate Simple Harmonic Motion?

[code.master]

I am working on a problem that has a few angles of approach. I am hoping to get at least that right before I waste too much more time. The problem is a slender, uniform, rigid rod is placed to pivot on its center, so that the rotation is taking place at the ends of the rod.

then a spring is attached to the bottom. small angle approximation allowed. initial displacement is $\theta$

We are to show $\frac{d^2\theta}{dt^2} = -\omega^2\theta$ and $T = \frac{2\pi}{\omega}$ and show how those imply SHM.

My problem is in setting up the equations. I was going to show that the rotational kinetic energy plus the spring potential energy was a constant... but I am guessing after working on it that approach isn't the best.. Any tips?

 

[quasar987]

Maybe it's just me but I don't understand the setup of the problem. Where is the spring exactly?!

 

[code.master]

um, let's describe it like this. a rod is vertical, and pinned at its midpoint. a spring is attached to its end and a wall or some arbitrary anchor. in my little drawing, the rod appears to be fixed to pivot on its endpoint, but let's assume there is a length above it of equal length, such that there is no gravity input here.

|\
|. \
| . \
| . . \---*spring*----| <- wall

dots are just to hold spacing.

 

[quasar987]

I assume the relaxed length of the spring is equal to the distance between the rod (when it is vertical) and the wall.

In this case, define a "coordinate system" like so

|\
|. \
| . \
| . . \---*spring*----| <- wall
|-x->

Theta is the angle between the rod and the vertical.

What is the force the spring exerts on the bottom of the rod? How can you use the small angle approx to express that in terms of theta? What torque does that force exerts about the CM of the rod (again, use small angle approx)? How is torque related to angular acceleration?

With the answer to all these questions, you should be good to go.