Rotational motion of pulleys influenced by gravity. Not sure how to set up.

[shanest]

I've attached the problem as a .jpg file. The mass obviously has potential energy. Is this simply converted to the kinetic energy of the mass plus the two kinetic energies of the pulleys? I also don't see how I would find the angular speed (probably don't even need to) of the pulleys. Cant simply due the gravity acceleration over the distance on the hanging mass since it has the tensions from the ropes. How do I set this problem up? Many thanks!

 

[AlephZero]

The mass obviously has potential energy. Is this simply converted to the kinetic energy of the mass plus the two kinetic energies of the pulleys? How do I set this problem up?

Yes, your statement about the energy is correct, so you answered your own question about how to set the problem up.

I also don't see how I would find the angular speed (probably don't even need to) of the pulleys.

You are given the radius and inertia of the pulleys. Because there is no slipping, the rope and pulley speeds are related by v = omega.r

 

[P3X-018]

You can also do this kinematically, where you can get the car's acceleration. Start by considering each body seperatly consider the forces on it. What forces are acting on the car? For the drum and the pulley, write down the "equation of motion" for both of them, that is, use the angular momentum theorem.
What is the relation between the angular acceleration of the pulley and the drum? And what is the connection between the acceleration of the car and the rotation? Is the tension force throughout the rope the same?
When you can get this done, you should get a certain number of equations with the same number of unknowns, that you should easily be able to solve.

 

[AlephZero]

You can also do this kinematically, where you can get the car's acceleration ... you should get a certain number of equations with the same number of unknowns, that you should easily be able to solve.

Yes you can do it that way.

The energy method gives you one equation, which you can solve for the one thing you are asked to find. That's got to be quicker and easier than setting up several equations involving new variables like tensions and accelerations, and then solving them all.

BTW if you learn about Lagrange's equations of motion in an advanced dynamics course, you will find you can also get the accelerations etc direct from the KE and PE of the system, without setting up all the other equations and solving them.