Simple Harmonic Motion on a Uniform Meter Stick

[NathanLeduc1]

Homework Statement

A uniform meter stick of mass M is pivoted on a hinge at one end and held horizontal by a spring with spring constant k attached at the other end. If the stick oscillates up and down slightly, what is its frequency?

Homework Equations

τ=rFsinθ
f=(1/2π)√(k/m)
F=kx
x=Acos(ωt)

The Attempt at a Solution

I'm really not sure how to get started on this one. If you could just provide me with a little start, I might be able to figure it out. Thanks.

The answer, according to the textbook, is (1/2π)sqrt(3k/m)

 

[tiny-tim]

Hi NathanLeduc1!

If you could just provide me with a little start, I might be able to figure it out.

Draw a force diagram for a small vertical displacement x, and find the force as a function of x.

(assume sinx = x)

 

[NathanLeduc1]

Ok, so I set up a force diagram and did the following work but I'm stuck again...

At equilibrium:
Ʃτ=Kx_ol-mg(l/2)=0

After it's been stretched:
Ʃτ=K(x+x_o)-mg(l/2)=Iα

This then simplifies to:
Iα=kx_ol

I wrote α as the second derivative of θ with respect to time but now I'm stuck. Where should I go from here? Thanks.

 

[tiny-tim]

(just got up :zzz:)

Iα=kx_ol

I wrote α as the second derivative of θ with respect to time but now I'm stuck. Where should I go from here? Thanks.

α = x''/l