# Cylinder sliding down a track

1. Oct 26, 2007

### delecticious

1. The problem statement, all variables and given/known data
A uniform solid cylinder (m=0.230 kg, of small radius) is at the top of a similar ramp, which has friction. The cylinder starts from rest and rolls down the ramp without sliding and goes around the loop. Find the speed of the cylinder at the top of the loop.

2. Relevant equations

PE - potential energy
KE - kinetic energy
KEr - rotational kinetic energy
I - moment of inertia
w - omega

KE0 + PE0 + KEr0 = KEf + PEf + KErf

3. The attempt at a solution

At first I was tripped up over the small radius part, until I realized that the radius would cancel out with final height of the potential energy and I deduced that the initial rotational and kinetic energies would cancel out to get something to this fashion:

mgh0 = 1/2mv^2 + mg2R + 1/2Iw^2
mgh0 = 1/2mv^2 + mg2R + 1/2I(v^2/r^2)
mgh0 = 1/2mv^2 + mg + 1/2Iv^2

I plugged and chugged my numbers to get a velocity of 7.18, but then I realized I forgot to incorporate the friction force, so how would that fit in to the mathematical equation?

2. Oct 26, 2007

### Staff: Mentor

OK.
Not sure what you did here. What happend to R and r?

I didn't do the calculation myself, but you need to double check your formulas. What did you use for I?

Since the cylinder rolls without slipping, no energy is lost to friction: Mechanical energy is conserved.

3. Oct 26, 2007

### delecticious

sorry, R=r so I should have had them in the same caps. Since they're the same I canceled them out

I = 1/2MR^2 <---- I looked it up in my book and it said the moment of inertia for solid cylinders is that
so does that mean the fact that there is friction is negligible in this situation?

4. Oct 26, 2007

### Staff: Mentor

R is the radius of the loop; r is the radius of the rolling cylinder. They are not the same. (And even if they were, how would they cancel?)

Good.

It's not that the friction is negligibly small, and thus can be ignored. Friction is essential--it's what makes the cylinder roll instead of slide. But from an energy point of view, the friction does no work so energy is still conserved. (So it has no effect on your energy equation.)

5. Oct 26, 2007

### delecticious

looks like goofed here. Ok so R stays, but r isn't given so somehow it drops out, but I'm not sure how it would.

6. Oct 26, 2007

### Staff: Mentor

Small r appears in the formula for I.

7. Oct 26, 2007

### delecticious

oh, no duh, thanks for the help