2 Block and a pulley(has mass) system

[okgo]

Homework Statement

http://i43.tinypic.com/352nlw2.jpg

I need help with the question 25. The three star (hardest) problem.

Homework Equations

well conservation of energy is
Mgh=.5Mv^2+.5Mv^2+.5(.5MR^2) ω^2

The Attempt at a Solution

Wouldn't increasing the Radius increase the Inertia by a factor of 4 and decreasing the mass reduce it only by a factor of 2. So the velocity should be less due to the increased inertia and the block would fall slower.

Anyone have any idea why the block would fall faster than the original?

 

[rl.bhat]

Since the single string is connecting the two masses, the acceleration of masses anfd the pulley must be the same. Let T1 be the tension in the horizontal segment of the string and T2 be the tension is the vertical segment of the string. Now draw the free body diagram. Applying Newton's second law, you have
T1 = ma ...(1)
Net torque τ = (T2 - T1)*R = I*α = 1/2*M*R^2*α. R*α = a. So
T2 - T1 = 1/2*M*a...(2)
mg - T2 = ma...(3)
From these equations find the expression for the acceleration a.
You can see that acceleration is independent of the radius of the pulley and it depends on the mass of the pulley.
In the acceleration expression the mass of the pulley appears in the denominator. So as mass decreases the acceleration increases.