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

To solve this problem, you will need to use the equation for angular acceleration, which is given by:

α = τ/I

where τ is the torque acting on the spool and I is the moment of inertia of the spool.

You can then use the equation for angular velocity, which is given by:

ω = αt

where α is the angular acceleration and t is the time.

Finally, you can use the equation for linear velocity, which is given by:

v = ωR

where ω is the angular velocity and R is the radius of the spool.

Using these equations, you can calculate the time it will take for the spool to run out of paper. First, calculate the angular acceleration using the equation α = τ/I. Then, calculate the angular velocity using the equation ω = αt. Finally, calculate the linear velocity using the equation v = ωR. The time it will take for the spool to run out of paper is equal to the linear velocity divided by the linear acceleration (v/a_tan).