A cylindrical disc of a gyroscope of mass `m=15kg` and radius `r=5.0cm` spins with an angular velocity `omega=330rad//s`. The distance between the bearings in which the axle of the disc is mounted is equal to `l=15cm`. The axle is forced to oscillate about a horizontal axis with a period `T=1.0s` and amplitude `varphi_m=20^@`. Find the maximum value of the gyropscopic forces exerted by the axle on the bearings.

Explanation

The moment of inertia is `1/2mr^2` and angular momentum is `1/2mr^2omega`. The axis oscillates about a horizontal axis making an instantaneous angle.
`varphi=varphi_m"sin"(2pit)/(T)`
This means that there is a variable precession with a rate of precession `(dvarphi)/(dt)`. The maximum value of this is `(2pivarphi_m)/(T)`. When the angle between the axle and the axis is at the maximum value, a torque `IomegaOmega`
`=1/2mr^2omega(2pivarphi_m)/(T)=(pimr^2omegavarphi_m)/(T)` acts on it.
The corresponding gyroscopic force will be `(pimr^2omegavarphi_m)/(lT)=90N`