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

 

[Nickie]

I would first start by defining the terms and variables in the problem. In this case, we have a uniform meter stick with a mass of M, a pivot point at one end, and a spring with a spring constant of k attached at the other end. The stick is oscillating up and down slightly, so we can assume that it is undergoing simple harmonic motion.

Next, I would use the given equations and information to solve for the frequency of the stick's oscillations. We can use the equation for torque (τ=rFsinθ) to find the restoring force of the spring (F=kx) and the distance from the pivot point to the center of mass of the stick (r=L/2, where L is the length of the stick). This will give us the equation τ=(L/2)kxsinθ.

Since the stick is held horizontal, we can assume that the angle θ is very small and can be approximated as sinθ≈θ. This allows us to simplify the equation to τ=(L/2)kxθ.

Next, we can use the equation for the angular frequency of simple harmonic motion (ω=(1/2π)√(k/m)) to find the frequency of the stick's oscillations. We know that the angular frequency is related to the linear frequency (f) by the equation ω=2πf, so we can rewrite the equation as f=(1/2π)√(k/m).

Finally, we can substitute in the value for x (x=Acos(ωt)) and solve for the frequency. This will give us the final equation, f=(1/2π)√(k/m)cos(ωt).

Plugging in the given values of k and m, we get a frequency of (1/2π)√(3k/m), which matches the answer given in the textbook.

In conclusion, by using the equations for torque and simple harmonic motion, we were able to solve for the frequency of the oscillations of the uniform meter stick. This frequency is dependent on the spring constant and the mass of the stick, and can be described by the equation f=(1/2π)√(k/m).