Two Rolling Objects Racing Down a Hill?

[dwangus]

Homework Statement

There are two objects rolling down a hill of incline theta, one is a sphere and one is a disk, each of equal radius and mass.
Which one gets down first and how much faster than the other?
What's the coefficient of static friction of the hill?

Homework Equations

Moment of inertia equations for disk and sphere.
Net Torque = Iα
α = a/r
Mgsinθ
Frictional Force
Net Force = ma

The Attempt at a Solution

Ok, so I left out information on purpose, because I just need the general solutions to work these things out.
I know that the sphere will obviously win because it has a lower coefficient of moment of inertia. And I know that for each object's moment of inertia will lead to different angular accelerations, which can lead you to understanding how fast each is individually going (alpha = a/r). And I also know that the net torque force is equal to I x alpha, and that the torques needed are mgsinθ and the force of friction... but in the end, I'm not quite sure how to put the frictional force and the gravitational force together to produce angular or linear acceleration, because the gravitational force acts through each object's center of mass, and the frictional force acts on the edge? I don't know how to reconcile the different axes...

And because of that, I absolutely do not know how to find the static friction of the cliff.
Please help? You can make up your own radii and masses and angles if you want, this is all purely theoretical anyways.

 

[haruspex]

Since it is a question about rolling, you don't really care about the frictional force beyond that it is enough to prevent sliding. So the easiest way is to take moments about a point for which the frictional force has no moment. Where would that be?

 

[dwangus]

The contact point of the object and the cliff?

 

[haruspex]

The contact point of the object and the cliff?

Yes.