We have three objects, a solid disk, a ring, and a solid sphere. If we release them from rest at the top of an incline, which object will win the race? Assume the objects roll down the ramp without slipping.
We can take the winner of this race, if there is one, and race it against a slippery block that slides down the ramp with negligible friction and see which one wins that race.
To analyze the rolling race, let's take an object with a mass M and a radius R, and a moment of inertia of cMR2. Consider the free-body diagram of such an object.
If there was no friction the object would slide down the ramp without rotating. Friction opposes this motion, so it must be directed up the slope. It's static friction because the object rolls without slipping.
Because the force of gravity and the normal force pass through the center of the object, the frictional force is the only force producing a torque about the center of the object - that's why the object rotates.
Let's analyze this situation from an energy perspective.
Because the object does not slip as it rolls, there is no loss of mechanical energy. As it rolls, the object is experiencing a combination of straight-line and rotational motion. Applying conservation of mechanical energy:
Ui + Ki = Uf + Kf
Take the bottom of the ramp to be the zero for the gravitational potential energy. If the object is released from a height h, the initial potential energy is mgh.
The initial kinetic energy is zero. The final kinetic energy is made up of translational and rotational kinetic energies.
Mgh = ½ Mv2 + ½ Iw2
Plugging in I = cMR2:
Mgh = ½ Mv2 + ½ cMR2 w2
Multiply both sides by 2, and cancel the mass (in other words mass doesn't matter):
2gh = v2 + cR2 w2
For rolling without slipping, the relationship between the velocity of the center-of-mass and the angular velocity is:
w | = |
|
, so: |
2gh | = | v2 | + |
|
The factors of R cancel, so the size of the object doesn't matter.
This gives:
2gh = v2 + c v2
Solving for v, | v | = | ( |
|
) | ½ |
So, the larger the value of c, the smaller the speed is. For our different objects we have:
ring | c = 1 |
disk | c = 1/2 |
solid sphere | c = 2/5 |
So, the sphere wins the race.