Falling Weight Attached to a Wheel and a Sphere

[Como Bluff]

How do I calculate the acceleration of the falling weight? It is hanging from a string which goes through a wheel, and is attached to a sphere with thin walls. The string doesn't stretch and the wheel and the sphere spin without friction.

The fact that the weight is connected to multiple masses is the problem. How does one calculate the net moment of inertia that is affecting the acceleration of the falling weight?

For the sake of simplicity, I left out the values given in the original statement of the problem; these are of course available upon request.

The moments of inertia for the objects in the problem:

J_sphere = $\frac{2}{3}$mr²
J_wheel = $\frac{1}{2}$mr²

Thanks!

 

[haruspex]

Deal with the tensions in the string, relating them to the torques.

 

[Como Bluff]

Thanks for quick reply haruspex! I don't quite get it yet. Could you elaborate your answer a little more?

 

[haruspex]

Thanks for quick reply haruspex! I don't quite get it yet. Could you elaborate your answer a little more?

Create unknowns for the tensions in the two sections of string. (They will not be the same.) Do the usual free body equations for each of the three massive components. You can relate all the accelerations and angular accelerations through the radii. You should have three equations with three unknowns (the two tensions and the acceleration).

 

[Como Bluff]

I got it now, thanks!

(I had also made the classic mistake of using the given diameter as the radius, which was needed to calculate the mass of the wheel from the moment of inertia..)