Question

A uniform sphere of mass $m=5.0kg$ and radius $R=6.0cm$ rotates with an angular velocity $\omega =1250rad/s$ about a horizontal axle passing through its centre and fixed on the mounting base by means of bearings. The distance between the bearings equals $l=15cm$. The base is set in rotation about a vertical axis with an angular velocity ${\omega }^{\prime }=5.0rad/s$. Find the magnitude of the gyroscopic forces in N.

Answer 1

The moment of inertia of the sphere is $\frac{2}{5}m{R}^2$
And hence the value of angular momentum is $\frac{2}{5}m{R}^2\omega$.
Since it precesses at speed ${\omega }^{\prime }$ the torque required is $\frac{2}{5}m{R}^2\omega {\omega }^{\prime }=F^{\prime }l$
So, $F^{\prime }=\frac{1}{l}\frac{2}{5}m{R}^2\omega {\omega }^{\prime }=300N$
(The force $F^{\prime }$ must be vertical.)