Solving for Acceleration in a Complex Pulley System

[TheLil'Turkey]

1. Homework Statement
What is the acceleration of m1? There is no friction so the pulleys don't rotate. The top rope is connected to the ceiling.

Homework Equations

See photo. I think my formula for the acceleration of M2 is only right if m1 = m2 and wrong in all other situations, but I'm not sure what the correct formula is.

The Attempt at a Solution

See photo.

 

[TheLil'Turkey]

In the equation for the acceleration of M2, I forgot to multiply it by g.
I uploaded a more recent photo of my attempted solution. It's in the OP above the original photo.

 

[haruspex]

You don't explain the basis for any of your equations, so you need to start at a more basic level. Assign an unknown to each tension and write out the ∑F_i=ma equation for each mass.
By the way, when a pulley is described as frictionless it nearly always means merely that there is no frictional torque at its axis. It does not usually mean there is no friction between the pulley and the rope. (If no friction there then it might as well just be a round mass, no need for it to be a pulley.) Thus, if the pulley has mass and a known radius then you should take into account its moment of inertia. However, if no radius is provided then clearly you have no way to take it into account.

 

[TheLil'Turkey]

There are no radii given. So it's best to assume it's truly frictionless. The problem is difficult enough without friction. If you or anyone else sees any mistakes in my equations I'd appreciate any help you could provide.

 

[haruspex]

There are no radii given. So it's best to assume it's truly frictionless. The problem is difficult enough without friction. If you or anyone else sees any mistakes in my equations I'd appreciate any help you could provide.

As I posted, your working is not immediately intelligible because you do not explain how you get any of your equations. Either explain them, or start at a more fundamental level with unknowns for tensions and a ∑F=ma equation for each mass.

You also need to define which way the various acceleration variables are defined. Are they all positive upwards? It doesn't look like it.

For what it's worth, your equations do not smell right to me. I would expect a m₁+m₂ term in your first equation.