Finding Angular Acceleration for a Rotating Wheel with a Hanging Mass

[flower76]

I'm trying to solve this problem, but keep going around and not getting the correct equation.

A wheel of radius R, mass M, and moment of inertia I is mounted on a frictionless horizontal axle. A light cord wrapped around the wheel supports a body of mass m. At time t=0, the body is let go and the wheel starts rotating.

Show that the angular acceleration of the wheel is:

$$\alpha\equiv\frac{g}{R+I/mR}$$

I know that $$I\alpha=TR$$ where T is the tension in the chord
and $$T=mg-ma$$, and $$a=R\alpha$$

But in trying to combine the equations something is not working out, I think I am missing something, but I can't figure out what. Please help.

 

[Doc Al]

I know that $$I\alpha=TR$$ where T is the tension in the chord
and $$T=mg-ma$$, and $$a=R\alpha$$

But in trying to combine the equations something is not working out, I think I am missing something, but I can't figure out what. Please help.

You're not missing anything. Do it step by step. Take the last equation, combine it with the second one (that is, eliminate "a" in the second equation). Then take your new expression for T in terms of alpha and combine it with the first equation to eliminate T. Now just solve for alpha.

 

[flower76]

Ok so I'm almost there, but when I try and solve for $$\alpha$$ things go a little off.

So I've got the equation down to $$\frac{I\alpha}{R}=mg-mR\alpha$$
Am I right so far? Now here is where my math skills fail me, and when I try and take $$\alpha$$ out of the equation, I definitely don't get the right answer.
Suggestions are welcome, thanks.

 

[flower76]

I've figured it out, thanks for the initial help!