What is the solution for the Pulley System Problem?

[Rugile]

Homework Statement

No friction, pulleys and also string are weightless, the tension through the string is distributed uniformly. Find the acceleration of the mass 3M (see image attached)

Homework Equations

II law of Newton

The Attempt at a Solution

Well I wrote such equations for all pulleys (counting pulleys from right to left):
1) M*a1 = T - Mg;
2) M*a2 = 2T - Mg;
4) M*a3 = 2T - Mg;
5) 3M*a = T - 3Mg.

Now we have 1 more unknowns than equations. Any ideas?

 

[BvU]

Can you make use of the fact that the rope has a fixed length ?

 

[Rugile]

Do you mean energy conservation?

Then, I guess, we could write eq.:
Initial state
E1 = 3Mgh + Mgh

Final state (when 3M descends h):
$E_2 = \frac{3Mv^2}{2} + \frac{Mv^2}{2} + 2*\frac{M\frac{v^2}{4}}{2} + 2Mgh + 2*Mgh/2$

Keeping in mind that the first mass M ascends h and has velocity v, second and third ascend h/2 and have velocity v/2 and the last one descends h and has velocity v.

Then

E2 = 9/4 Mv^2 + 3Mgh

Is that true? Then the rest is clear :)

 

[phoenixXL]

No, I guess BvU is saying to use string constraints.
A detailed explanation for proceeding, if you don't know what string contraint is, can be found
Part A
Part B

 

[Rugile]

Thanks, that's extremely helpful!
Had no idea about such thing!