In the arrangement shown in figure the sleeve `M` of mass `m=0.20kg ` is fixed between two identical springs whose combined stiffness is equal to `x=20 N//m`. The sleeve can slide without friction over a horizontal bar `AB`. The arrrangement rotates with a constant angular velocity `omega=4.4 rad//s` about a vertical axis passing through the middle of the bar . Find the period of small oscillations of the sleeve. At what values of `omega` will there be no oscillations of the sleeve ?

Explanation

Obviously the sleeve performs small oscillations in the frame of rotating rod. In the rod's frame let us depict the forces acting on the sleeve along the length of the rod while the sleeve is at a small distance `x` towards right from its equilibrium position. The free body diagram of block does not contain Coriolis force, because it is perpendiucluar to thelength of the rod.
From `F_(x)=mx_(x)` for the sleeve in the frame of rod
`-kx + m omega^(2)=mddot(x)`
or, `ddot(x)=-((k)/(m)-omega^(2))x`
Thus the sought time period
`T=(2pi)/(sqrt((k)/(m)-omega^(2)))=0.7s`
It is obvious from Eqn `(1)` that the sleeve will not perform small oscillations if
`omegagt = sqrt((k)/(m))=10 rod //s.`