We release a solid disk, a ring, and a solid sphere from rest at the top of an incline. The objects roll without slipping. Which object will win the race?
Consider an object with mass M, radius R, and moment of inertia cMR2. We'll analyze the race from two different perspectives.
Because each object does not slip as it rolls, there is no loss of mechanical energy by friction. Thus: Ui + Ki = Uf + Kf
Define h = 0 at the bottom of the ramp; thus Uf = 0. If the object is released from a height h, Ui = mgh.
The initial kinetic energy is zero. The final kinetic energy consists of translational and rotational parts: Kf = ½ Mv2 + ½ Iω2
Apply energy conservation: Mgh = ½ Mv2 + ½ Iω2
Using I = c MR2 gives:
Mgh =½ Mv2 + ½
c MR2 ω2
2gh = v2 + cR2 ω2
Here c is an arbitrary constant.
For rolling without slipping, the relation between center-of-mass velocity and angular velocity is ω = v/R so:
The factors of R cancel, so size doesn't matter. We obtain: 2gh = v2 + c v2
|Solving for v,||v =|| |
So smaller c gives a larger speed. For our objects we have:
|ring||c = 1|
|disk||c = 1/2|
|solid sphere||c = 2/5|
The sphere wins!
The force of gravity and the normal force pass through the center of the object and produce no torque about the center-of-mass. The frictional force is the only force that produces a torque about the center-of-mass.
Apply DID TASC:
Step 3 -- Draw all forces: There are two
forces parallel to the slope.
the component of gravity acting downslope mg sin θ;
the static friction force, fs, up the slope.
Step 5 -- Apply Newton's 2nd Law: ΣF = Ma Σ τ = I α
The force equation is (+x = downslope): Mg sin
θ - fs = Ma
The torque equation about the CM is (+z = out): fsR = Iα
For rolling without slipping, α = a/R, and we also use I = c M R2 so that the torque equation becomes: fs R = cMR2 a / R.
The factors of R cancel, leaving: fs = cMa
Substituting this into the force equation gives: Mg sin θ - cMa = Ma
Solving for the acceleration gives:
Both the acceleration and the velocity are reduced by a factor of 1+c
compared to the case of no friction. Thus an object that slides without
friction beats any rolling object. For the rolling race, the object with the
smallest c in I = cMR2 wins, independent of the mass and the
Question: what attribute produces the smallest value of c in I = c M
r2 for fixed mass and fixed radius?