A roll of toilet paper (a partially hollow cylinder with R_2 = 7.5cm, M = 300g, and I = 9.0 x 10^-4 kg m^2) is mounted on a mass-less axle along its central axis. Te roll is initially at rest. Then, at t=0, a child grabs the end of the roll and starts running, pulling the paper off the roll at a constant linear acceleration a_tan = 0.35 m/s^2.

Question

*Assume that the roll has uniform density. Throughout this question, assume that both M and R_2 remain constant, even though paper is unspooling from the roll.*

QUESTION: 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?

------- Where do I begin with this problem? -----
From the other parts of the problem I found out that the inner radius, R_1 is 0.019m (1.9 cm) and that the magnitude of the torque acting on the spool is .0042 N m (not sure if I did my work here correctly.) What equations would I have to use to solve the problem?

Answer 1

The paper runs out when the child has run 40 meters. The rest is irrelevant.

40 = (1/2) at^2

80 = .35 t^2

t = sqrt (80/.35)

Answer 2

no