Why Do Two Falling Objects Have the Same Acceleration in This Physics Problem?

[wu7]

Homework Statement

One pulley, on one side we have a block with mass m.
On the other side we have a cylinder with mass m.
Cylinder (radius R) has unlimited string (massless, negligible thickness, no slippage)

So you can imagine two blocks falling as more string unravels from the cylinder.

I am told that these two objects have the same acceleration downwards. Why?

Homework Equations

F=ma
ma=mg-T

torque? T*R=I*$\alpha$

The Attempt at a Solution

The problem made it sound like this was a quick, obvious argument.
I proceeded to a longer argument, finding that the accelerations of both masses were (2/3)g downward:

starting with the cylinder-
TR=I*$\alpha$
T*R² = (1/2)M*R² * a
Finding T= (1/2)ma, then plugging T back into the standard F=ma equations.
Then I find the acceleration of the block to be (2/3)g as well.

Is this even right? If it is, was there a way to show that the objects' accelerations were the same without finding the actual accelerations?

The reason I don't think this is right is because I did the problem using conservation of energy using the assumption that the objects fell down at the same acceleration.
With conservation of energy I got acceleration=(1/2)g (conserving KE, PE, angular KE)

 

[gDavidov]

I think you don't need to use torque.
By F= m * a(center of mass) you get that their accel will only depend on forces applied to them. In both cases block and cylinder you have F = T - mg.
Then => m * aBlock = m * aCylinder => aBlock = aCylinder