Torque question.?

Question

a gyroscope consists of a metal disc of moment of inertia
I = 4.5 x 10^-5 kg m^2, mounted on a light axle of radius 2 mm
which rotates freely within a frame

I set the gyroscope in motion by pulling a 30 cm length of string, which I wound around the axle. I now apply a force of
10 N (in a direction perpendicular to the axle) to the string as it unwinds. Friction / air resistance negligible.

1. Calculate final angular speed & magnitude of angular momentum of the gyroscope

2. How long would it take me to pull out the 30 cm of string.

I calculated the work done ( 3 Nm) and torque (0.02 Nm)

Answer 1

1. KE = Work = 10 N * 0.3 m = 10 J
KE = Iw^2/2
Ang. rate w = sqrt(2KE/I) = 666.7 rad/s
Ang, mom. H = Iw = 0.03 N-m-s
2. Torque T = F*r = 10 * 0.002 = 0.02 N-m
Time t = H/T = 0.03/0.02 = 1.5 s