Question

A student holds one end of a thread, which is wrapped around a cylindrical spool, as shown above. The student then drops the spool from a height h above the floor, and the thread unwinds as it falls. The spool has a mass M and a radius R, and the thread has negligible mass. The spool can be approximated as a solid cylinder of moment of inertia I = 1 MR2. Express your answers in terms of M, R, h, and fundamental constants. At time t = 0, the spinning spool lands on the floor without bouncing and comes free from the thread. It continues to spin, but slips on the floor's surface while doing so. Assume a constant coefficient of sliding friction m. (c) Calculate the angular velocity of the spool as a function of time t.

Answer 1

by energy conservation we know that

KE or rotation + KE of translation = gravitational PE

now we have

$\frac{1}{2}I\omega^2 + \frac{1}{2}mv^2 = mgH$

also we know that

$v = R\omega$

now we have

$\frac{1}{2}(\frac{1}{2}mR^2)\omega^2 + \frac{1}{2}m(R\omega)^2 = mgH$

$\frac{3}{4}mR^2\omega^2 = mgH$

$\omega = \sqrt{\frac{4gH}{3}}/R$

now when it is rolling on ground the torque acting on it due to friction force is given by

$\tau = R F_f$

$\tau = \mu mg R$

$\alpha = \frac{\mu mg R}{\frac{1}{2}mR^2}$

$\alpha = \frac{2 \mu g}{R}$

now angular speed at any time is given as

$\omega = \omega_i + \alpha t$

$\omega = \sqrt{\frac{4gH}{3}}/R -\frac{2 \mu g}{R} t$

so above is the angular speed in terms of time "t"