# Oscillating system (small board on a halfpipe)

## Main Question or Discussion Point

Hello there!

I have a situation as follows:

[PLAIN]http://img821.imageshack.us/img821/4026/kuglananagibu.gif [Broken]

I have to find the period of oscillation of the system. I've know how to set the equations for when solving the equations with forces, but am lost when trying to solve it with energy.

There is no friction between any of the surfaces.

My assumption is that there should be three energies interchanging here.

Rotationl energy of m (around the center of the circle with radius R)
$$W_{rot}=1/2 mR^2 \omega$$

And translational energies of M and m.
$$W_{trans}=1/2 Mv^2 + 1/2m v^2\omega$$

Any ideas where to go form here?

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Delta2
Homework Helper
Gold Member
$$mgR(1-cos(\phi))+\frac{1}{2}mR^2{\omega}^2=c_0$$ with $$\omega=\frac{d\phi}{dt}$$.

You also might find usefull the trigonometric identity $$cos(\phi)=1-2sin^2(\frac{\phi}{2})$$. If you consider small amplitudes then you can also take the approximation $$sin^2(\frac{\phi}{2})=(\frac{\phi}{2})^2$$. If you use both you ll end up with a differential equation that doesnt look simple , yet it has solutions of the form $$\phi(t)=asin(bt)$$

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Spinnor
Gold Member
Why do you include the kinetic energy of the half-pipe, does it move?

What you have here is similar to a physical pendulum? Do a Google search,

"period of a pendulum large amplitude oscillation" or something like that.

I should have metnion more explicitely, the halfpipe has a constant mass M and lies on a frictionless surface. Therefor is also oscillates.

Spinnor
Gold Member

I should have metnion more explicitely, the halfpipe has a constant mass M and lies on a frictionless surface. Therefor is also oscillates.

Can you assume small amplitude motion?

Spinnor
Gold Member
Can you write down the Lagrangian for this system, T - V, and then solve the Lagrange equations of motion? What text book are you using?

Delta2
Homework Helper
Gold Member

I should have metnion more explicitely, the halfpipe has a constant mass M and lies on a frictionless surface. Therefor is also oscillates.
Hm now it becomes really interesting. You have to add $$\frac{1}{2}(M+m)v^2$$ to the left hand side of the equation(i guess M oscilates horizontaly so his potential energy doesnt change). But apparently we need one more equation and i can think only one that uses forces though. If $$F_n$$ is the force between m and M then $$F_n-mgcos(\phi)=m{\omega}^2R$$ and $$F_nsin(\phi)=M\frac{dv}{dt}$$

Can you assume small amplitude motion?
Yes.

Spinnor said:
Can you write down the Lagrangian for this system, T - V, and then solve the Lagrange equations of motion? What text book are you using?
I'm not on Lagrangian mechanics yet, it's first year physics.

Hm now it becomes really interesting. You have to add $$\frac{1}{2}(M+m)v^2$$ to the left hand side of the equation(i guess M oscilates horizontaly so his potential energy doesnt change). But apparently we need one more equation and i can think only one that uses forces though. If $$F_n$$ is the force between m and M then $$F_n-mgcos(\phi)=m{\omega}^2R$$ and $$F_nsin(\phi)=M\frac{dv}{dt}$$
Shouldn't there also be a force component of the system force on the board? (Due to the acceleration of the halfpipe).

I don't see how including forces into energy equations could help. Maybe they could be connected by

Wk = x F (for the kinetic energy of the halfpipe)

it's just, i've never seen a scenario in which those two interact. Could there a purely energy solution be found by using the center of mass? Or perhaps that the center of mass doesn't move in the x direction?