# Atwood Machine, deriving equation for acceleration

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1. Sep 3, 2015

### mmoe

1. The problem statement, all variables and given/known data
I still dont got the hang on Atwood Machines, and i dont know if im 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 im 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 atr.

a) Show that atr = (a2 + a3)/2

b) find the masses' acceleration

2. Relevant equations
F=ma

3. 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.

ym1 = 1/2*ym2
ym1 = 1/2*y2m
ym2 = y2m

Hence ym1 = (ym2 + y2m)/2

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

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 dont know where to start here. Since a2 = a3, you only have to solve for a1 and a2 (correct me if im wrong). But how would i go forward solving (1) and (2) with (4). I dont know if im overthinking this way to much or if i have done something wrong, so i come to you kindly asking for directions.

Thanks :)

2. Sep 4, 2015

### 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?