Look her physics people plz?!?

Question

I did this experiment in class. It was a simple experiment. My teacher had a wooden ramp leaning up against the wall. He had a wooden disk and a metal ring. By ring, I mean it had no inside, just like a ring. By disk, I mean it had an inside to it. Both the wooden disk and the metal ring had the same mass, same diameter, same size and same shape. They were released down the ramp at the same time and the wooden disk always made it down first. My question...Why does the disk always beat the ring? Explain and show a mathematical proof why.

I really have no idea as to why except that there must be a force acting on them?!? I have no clue where to start on the mathematical proof....can someone please help me asap? Thank you tons if you can help.

Answer 1

for the metal ring, the mass is around the outside of the object, so it takes more to get going. the disc has its mass spread out through the whole thing, which makes it easier. just like if you spin around in a circle with your arms in, you spin faster than with your arms out.

Answer 2

The weights of the disk and ring are equal. When they're on the ramp, the weights can be resolved into downslope and perpendicular-to-the-slope components. The downslope component is opposed by friction in both cases and that provides a turning torque.

Since:
the downslope components of both's weight are equal, the opposing friction is equal,
and the diameter was the same for each, therefore
the torque, T = F*LeverArm, will be equal.

Rotational acceleration, alpha, is given by
T = I*alpha
where I is the rotational inertia. Note: this is the rotational equivalent of the equation F = m*a.

For a disk
I = M*R^2 / 2
For a ring
I = M*R^2

Therefore with Torques the same for both, the one with 1/2 the I, will have 2X the rotational acceleration.