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## Homework Statement

A point mass m is attached to the rim of an otherwise uniform solid disk of mass M and radius R (in the diagram mass m is in contact with the ground). The disk is rolled slightly away from its equilibrium position and released. It rolls back and forth without slipping. Show that the period of this motion is given by

[tex] T=2\pi \sqrt{\frac{3MR}{2mg}}[/tex]

## Homework Equations

[tex]I=\frac{1}{2}MR^2[/tex]

[tex]I \frac{d^2 \theta}{dt^2}=-\kappa \theta[/tex]

[tex]\omega=\sqrt{\frac{\kappa}{I}}[/tex]

[tex]T=\frac{2\pi}{\omega}[/tex]

[tex]\alpha = \frac{d^2\theta}{dt^2}[/tex]

## The Attempt at a Solution

My guess would have been

[tex]I=\frac{1}{2}MR^2 + mR^2[/tex]

and then

[tex]I\alpha = -mgR\sin\theta \approx -mgR\theta[/tex]

now we have kappa and I so we can pop that into solve for T which gives a big mess. This problem is a bit tricky for me as I am trying to look for a stationary axis to take torques about and not sure about how to apply the parallel axis theorem with the point mass (do I even need to use it?). I tried calculating the centre of mass of the disk just for kicks and it turned out to be a point 2/3R from the centre (using Centre of Mass = [itex]\int r dm[/itex]) but it should be in the centre...thats another story.

It seems there must be a pretty easy way to solve it but I can't see it.