# Angular acceleration of a wheel w/string on inner hub?

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1. Feb 6, 2017

1. The problem statement, all variables and given/known data
“A bicycle wheel is mounted as in the lab and as shown to the right. This wheel has a mass of 6.55 kg, a radius of R = 38.0 cm, and is in the shape of a ring. A mass M = 1.85 kg is attached to the end of a string which is wrapped around an inner hub which has a radius r = 5.40 cm. Initially, the mass M is at a distance h = 72.0 cm above the floor. Assume friction is negligible.”

a. What will be the resulting angular acceleration of this wheel?

b. How long will it take for mass M to reach the floor?

c. What will be the total angular displacement of the wheel during the time in which the mass M is falling to the floor?

d. How much work was done on the wheel by the external torque as the mass M falls to the floor?

e. What will be the angular kinetic energy of this wheel just as mass M reaches the floor?

2. Relevant equations
τ = Iα

F=ma

a=αr

I=mr^2

3. The attempt at a solution

I thought that Tr = Iα, and since T is mg-ma, I rewrote it as r(mg-ma)=(mr^2)α. Then, I tried to just divide r(mg-ma) by mr^2 to get α.

My process went like this:

[r(mg-ma)] / (mr^2) = α

(mgr-mar) / (mr^2) = α

mr(g-a) / (mr^2) = α

(g-a)/r = α

But that doesn’t seem to be helping me very much, as I don’t know a.

Since the rest of the questions seem to be dependent upon the answer to Part A, I didn’t know whether or not to try them.

Thanks so much to anyone who helps. I think once I get Part A I’ll be able to do the rest

2. Feb 6, 2017

### haruspex

Can you think of another relationship between a and α?

3. Feb 6, 2017

I've only ever been taught that angular acceleration is equal to the linear acceleration divided by the radius. I do know that α=Δω/Δt, and because ω is related to velocity, the α=a/r thing can be derived. Am I missing something important?

4. Feb 6, 2017

### haruspex

So use that. But there are some mistakes in your work so far. There are two different masses and two different radii. You need to be careful not to confuse them.

5. Feb 6, 2017