How Do You Solve a Double Atwood's Machine Problem?

[struggles]

Homework Statement

Masses m₁ and m₂ are connected by a light during A over a light frictionless pulley B. The axel of pulley B is connected by a light string C over a second light frictionless pulley D to a mass m₃. Pulley D is attached o the ceiling. The system is released from rest.
In terms of m₁, m₂, m₃ and g what are
a) the acceleration of the block m₃
b) the acceleration of pulley B
c) the acceleration of block m₁ and m₂
d) The tension in the string A
e) The tension in the string C

Homework Equations

F = ma

The Attempt at a Solution

a) As the strings are weightless the tension either side of the pulley will be the same.
I came up with equations
T_A - m₁g = m₁a₁
T_A - m₂g = -m₂a₁
T_B - m₃g = m₃a₂
T_B - (m₁ + m₂)g = -(m₁ + m₂)a₂
T_B = 2T_A

Rearranging the first 2 equations i got
T_A = (2m₁m₂g)/(m₁ + m₂)

I then substituted this into equation 3 to get
(4m₁m₂g)/(m₁ + m₂) - m₃g = m₃a
which when i rearrange goes
(4m₁m₂g - m₁m₃g - m₂m₃g)/(m₁m₃ + m₂m₃) = a

However this is not the answer stated in my textbook and I'm not sure where I've gone wrong.
Any help would be much appreciated!

 

[Jilang]

Your first equations would be right only if the pulley B were fixed.

 

[struggles]

Your first equations would be right only if the pulley B were fixed.

So would it be T_A - m₁g = m₁(a₁ + a₂) as it is accelerating not only on its own pulley system but also has the acceleration of pulley B?
This would also change equation 2 to T₁ - m₂g = -m₂(a₁ + a₂)

 

[TSny]

If a₁, a₂, and a₃ are accelerations measured relative to the earth, then equation 1 is OK. The next three need modification. You are right that you are going to need to think about relative accelerations.