A roll of toilet paper (a partially hollow cylinder with R 2 = 7.5cm, M = 300. g

Question

A roll of toilet paper (a partially hollow cylinder with R2 = 7.5cm, M = 300. g, and I = 9.0 x 10-4 kg m2 ) is mounted on a massless axle along its central exis. The roll is initially at rest. Then, at t = 0, a rascally child grabs the end of the roll and starts running, pulling the paper off the roll at a constant linear acceleration atan = 0.35 m/s2.

(Assume that the roll has a uniform density. Throughout this question, assume that both M and R2 remain constant, even though paper is unspooling from the roll.)

a. What is the spool's linear radius, R1?

b. What is the magnitude of the torque acting on the spool?

c. At what time does the spool reach a rotation rate of 400. rpm?

d. The spool holds only 40. m of paper. If the child maintains the same constant acceleration, at what time will the spool run out of paper? (Even though paper is unspooling from the roll, assume that R2 and M remain constant).

e. At the moment the spool runs out of paper, how many revolutions has it executed? (Even though paper is unspooling from the roll, assume that R2 and M remain constant).

Explanation / Answer

part a) using the following equation for Moment of inertia for various bodies (e) cylinder find that:

I = (1/2)M(R_1^2 + R_2^2)

solve for R_1
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part b)

a_tan = radius * angular acceleration
solve for angular acceleration
angular acceleration = (change in angular velocity) / (change in time)
solve for time
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