Rolling Paper: Analyzing Motion and Energy

[montreal1775]

Homework Statement

A large, cylindrical roll of paper of initial radius R lies on a long, horizontal surface with the open end of the paper nailed to the surface. The roll is given a slight shove (initial velocity is negligible) and begins to unroll. Determine the speed of the center of mass of the roll when its radius has diminished to r. Assume the roll has uniform density.

Homework Equations

$$K_i + U_i=K_f+U_f$$

The Attempt at a Solution

I have set up the problem using conservation of energy:

$$M_g R = (1/2) m v^2 + (1/2) I \omega^2 + m g r$$

I think this is the proper way to set it up, but I don't know how to find the relationship between the initial mass, final mass, and the radius.

 

[Dorothy Weglend]

That looks good to me.

For your question, while the masses will change, the density of the paper will remain the same. Use that fact.

Dorothy