Aceleration & tension in a system involving pulley

[jb007]

1. Homework Statement
I am stuck on this problem involving tension, acceleration of a system, friction and a pulley. I have an idea of how to find the acceleration of the system neglecting the mass of the pulley, but it is not the solution. How does adding a frictionless pulley in the system change its acceleration?

Homework Equations

F=ma
torque = I(alpha)
rotational equations?

The Attempt at a Solution

For the tension T₁ from the pulley disk downwards to mass m, the tension is mg.
For the tension T₂ from the mass M rightward to the pulley disk, would the tension be equal to T₁?

I drew FBD's for the two masses.
For M, by applying Newton's 2nd Law, F_net=Ma: T₂-f = Ma
For m, applying F=ma: mg-T₁ = ma

I know the acceleration of the system should be equal. But how do I incorporate the pulley disk in the system to find the acceleration of the system?

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[Simon Bridge]

You draw free body diagrams for both blocks and the pulley ... hint: what sort of motion does the pulley undergo?

 

[jb007]

So the pulley disk is rotating. The string wraps around the top of the disk, and it rotates in the same direction as the acceleration of the two blocks (the same way the string is moving). Would this be a torque force acting on pulley disk? Because the force on the disk by the string is perpendicular to the radius?

 

[Simon Bridge]

The disk must be accelerating (unless the rope is slipping) so there must be an unbalanced torque in the direction of the acceleration.
The free body diagram for the pulley has two forces on it - from each straight section of string.
The forces produce opposing torques. The tension on either side of the pulley will be different.

 

[jb007]

So the tension on the left side of the pulley would be the tension caused by the frictional force acting on M, and the tension on the right side of the pulley would be caused by the weight of m? (They are in opposite directions)

 

[Simon Bridge]

As far as the free body diagram of the pulley is concerned, there is no mass m or mass M.
Just write in T1 and T2 for the two tensions ... they come from the interactions between parts of the overall system.