Atwood's Machine, massive pulley and upwards force acting upon it.

[DavidDante]

Homework Statement

The usual configuration of an Atwood's Machine.
Two masses: m1, m2
Accelerations (unknown): a1, a2, ap (Pulley)
Rope Tension: T (Unknown)
BUT:
Massive pulley: mp
Upwards force acting on pulley (i.e the pulley is not in equilibrium): F

NOTE: Consider that all the pulley's mass in concentrated at the center of the pulley, so as not to consider the moment of inertia. (Is this ok?)

Question: Find the tension T, and the accelerations a1, a2.

Homework Equations

Newton's Second Law

The Attempt at a Solution

So this is what I tried:
Analyzing the pulley.
F - 2T - mp . g = mp . ap.........(I)

Analyzing the whole System as a unit:
F - (mp + m1 + m2) . g = (mp + m1 + m2) . ap....(II)

Analyzing m1:
T - m1 . g = m1 . a1..........(III)

From (II) I derived ap in terms of known constants, then I replaced ap (in terms of the constants) in (I) and I found T in terms of known constants. Finally I replaced the equation for T in (III).


So this is what I've got. Did I miss anything? Any feedback would be greatly appreciated.

Thanks-

 

[Simon Bridge]

Consider that all the pulley's mass in concentrated at the center of the pulley, so as not to consider the moment of inertia. (Is this ok?)

That is only OK if the operation of the machine means that the pulley does not rotate.

The approach you want is to draw three free-body diagrams, one for each mass and another for the pulley. You need to relate the tensions to the torques on the pulley.