A Race: Rolling Down a Ramp

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.

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 the problem from a force and torque perspective. We'll analyze it again later on using energy.

Forces and Torques

The free-body diagram of the object shows two forces parallel to the slope. If the angle of the incline is θ, the component of the force of gravity acting down the slope is Mg sin(θ).
f_s, the force of static friction, acts up the slope.

Summing forces in this direction, with positive down the slope, gives:

Mgsin(θ) - f_s = Ma

With both the normal force and the force of gravity passing through the center-of-mass of the object, summing torques about the center of mass gives:

f_sR = I α
I = cMR² and, for rolling without slipping, α =
a

R

The torque equation becomes:
f_s R =
cMR² a

R

The factors of R cancel, leaving:

f_s = cMa

Substituting this into the force equation gives:

Mgsin(θ) - cMa = Ma

The factors of mass cancel. Solving the equation for acceleration gives:
a =
g sin(θ)

1 + c

When starting from rest, v = at, so both acceleration and velocity are reduced by a factor of 1+c from what they would be if the object slid down the ramp with no friction.

Whichever way you analyze it, the object with the smallest value of c in I = cMR² wins the race. The mass and radius don't make any difference - they canceled out in the equations.