Help finding the work done to change the radius of a circle?

Question

A small block of mass (so it may be treated as a point mass) 0.91 kg slides without friction on a horizontal table. Initially it moves in a circle of radius r0 = 0.630 m with a speed 1.8 m/s. It is held in its path by a string that passes through a small hole at the center of the circle. The string is then pulled down a distance d = r0 - r1 = 0.215 m, leaving it at a radius r = r1 = 0.415 m. It is pulled so slowly that the object continues to move in a circle of continually decreasing radius. How much work was done by the force to change the radius from 0.630 m to 0.415 m?

Picture to go with problem: http://i284.photobucket.com/albums/ll28/bathtub2008/showmepl-7.gif

W = ____ J

Answer 1

Conservation of momentum:

I0ω0 = I1ω1

I1ω1 = mr0^2*1.8/r0 = mr0*1.8

I2ω2 = mr1^2*v1/r1 = mr1v1

v1 = mr0*1.8/(mr1) = 1.8r0/r1

Energy difference = m/2*(v1^2-1.8^2)

From conservation of energy, this difference is the work done by the string tension and the radius change.