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^'=5.0rad//s`. Find the modulus and direction of the gyroscopic forces.

Explanation

The moment of inertia of the sphere is `2/5mR^2` and hence the value of angular momentum is `2/5mR^2omega`. Since it precesses at speed `omega^'` the torque required is
`2/5mR^2omegaomega^'=F^'l`
So, `F^'=2/5mR^2omegaomega^'//l=300N`
(The force `F^'` must be vertical.)