Atwood Machine, deriving equation for acceleration

[mmoe]

Homework Statement

I still don't got the hang on Atwood Machines, and i don't know if I am wrong here or just overthinking it. But i got this one exercise where i am kind of stuck in two of the sub-exercises. So I am hoping there is someone here who could guide me in the right direction.Atwood Machine ( http://imgur.com/Dzvt4HR ), with massless strings and pulleys and no friction.

Let the acceleration of the top mass be a₁ and the acceleration for the bottom two a₂ and a₃. Let the movable pulleys have the acceleration a_tr.

a) Show that a_tr = (a₂ + a₃)/2

b) find the masses' acceleration

Homework Equations

F=ma

The Attempt at a Solution

a) I can see that if i move the top mass a distance y, conservation of string tells me that the movable pulleys will move 1/2y.

y_m1 = 1/2*y_m2
y_m1 = 1/2*y_2m
y_m2 = y_2m

Hence y_m1 = (y_m2 + y_2m)/2

Two time derivatives gives a₁ = (a₂ + a₃)/2 = a_tr

It gives the right answer, but i got a feeling that I've done something that should be quite obvious wrong. b) I find the F=ma for each mass

T-mg = ma₁ (1)
T-mg = ma₂ (2)
t-2mg = ma₃ (3)
i also got these equations from the conservation of string:
a₁=1/2a₂ (4) && a₁=1/2a₃ (5) && a₂=a₃ (6)

Combining (4) and (5) with equations (1),(2) & (3), should give me a₁,a₂,a₃ and T.

The thing is, i really don't know where to start here. Since a2 = a3, you only have to solve for a1 and a2 (correct me if I am wrong). But how would i go forward solving (1) and (2) with (4). I don't know if I am overthinking this way to much or if i have done something wrong, so i come to you kindly asking for directions.

Thanks :)

 

[haruspex]

I don't understand some of your equations. What is ##y_{m1}=\frac 12 y_{m2}## saying? Are these positions or velocities or accelerations? Why would it be true?