Rotational dynamics Find acceleration

[ForceFail]

Homework Statement

http://imgur.com/eYoa8Ut

Homework Equations

Torque=rt=I(alpha)
t-mg=-ma
alpha = a/r

The Attempt at a Solution

t-mg=-ma is the equation for the hanging mass.
To find how much this turns the small wheel,
rt=I(alpha) I and r are known
rt=Ia/r

I'm not sure what to do from here. If I try and solve for a and t now, it gives an incorrect answer. I think this is due to me not doing anything with the larger wheel yet, but I'm not sure what to do with it. I know that the tension in the upper and lower part of the belt are not equal, and both produce torque of their own. Tupper should produce a clockwise moment, Tlower should produce a counterclockwise moment.

 

[haruspex]

Tupper should produce a clockwise moment, Tlower should produce a counterclockwise moment.

Quite so. So include these in your torque equation for the smaller wheel and write down the torque equation for the larger wheel.

 

[ForceFail]

So for the smaller wheel:
tr+(Tl)R-(Tu)R=I(alpha)
Larger wheel gives:
(Tu)(Rlw)-(Tl)(Rlw)=I(alpha)

So now I have 2 equations and four unknowns. t,(Tl),(Tu),(alpha)
I can write t=mg-ma and alpha as a/r, Does alpha = a/r for the small wheel use the smaller or larger radius? Since its a tangential I would assume the larger.
That gives
(mg-ma)r+(Tl)R-(Tu)R=Ia/R
and
(Tu)(Rlw)-(Tl)(Rlw)=Ia/(Rlw)

If this is right I'm now down to 3 unknowns and only 2 equations.

 

[haruspex]

So for the smaller wheel:
tr+(Tl)R-(Tu)R=I(alpha)
Larger wheel gives:
(Tu)(Rlw)-(Tl)(Rlw)=I(alpha)

So now I have 2 equations and four unknowns. t,(Tl),(Tu),(alpha)
I can write t=mg-ma and alpha as a/r, Does alpha = a/r for the small wheel use the smaller or larger radius? Since its a tangential I would assume the larger.
That gives
(mg-ma)r+(Tl)R-(Tu)R=Ia/R
and
(Tu)(Rlw)-(Tl)(Rlw)=Ia/(Rlw)

If this is right I'm now down to 3 unknowns and only 2 equations.

Look carefully at how Tu and Tl appear in these equations. Notice anything?

 

[ForceFail]

Look carefully at how Tu and Tl appear in these equations. Notice anything?

I see, multiplied equation 1 by R(lw) and equation 2 by R and added them. Then Tu and Tl cancel and you can solve for a. Got the right answer Thanks.

 

[haruspex]

I see, multiplied equation 1 by R(lw) and equation 2 by R and added them. Then Tu and Tl cancel and you can solve for a. Got the right answer Thanks.

Good.
Note what this means physically. Suppose the belt is elastic. Making it tighter will increase both tensions equally, but won't affect the answer to the question.