A horizontally oriented uniform disc of mass `M` and radius R rotates freely about a stationary vertical axis passing through its centre. The disc has a radial guide along which can slide without friction a small body of mass `m`. A light thread running down through the hollow axle of the disc is tied to the body initially the body was located at the edge of the disc and the whole system rotated with ann angular velocity `omega_(0)`. Then by means of a force `F` applied to the lower and of the thread the body was slowly pulled to the rotation axis. find:
(a). The angular velocity of the system in its final state.
(b). The work performed by the force F.

Explanation

(a). `I_(1)omega_(2)=I_(2)omega_(2)`
`thereforeomega_(2)=((I_(1))/(I_(2)))omega_(1)=(((1)/(2)MR^(2)+mR^(2)))/(((1)/(2)mR^(2)))omega_(0)`
`=(1+(2m)/(M))omega_(0)`
(b). `W=K_(f)-K_(i)`
`=(1)/(2)I_(2)omega_(2)^(2)-(1)/(2)I_(1)omega_(1)^(2)`
`=(1)/(2)[(1)/(2)MR^(2)][1+(2m)/(M)]^(2)omega_(0)^(2)-(1)/(2)`
`[(1)/(2)MR^(2)+mR^(2)]omega_(0)^(2)`
`=(1)/(4)MR^(2)(1+(2m)/(M))omega_(0)^(2)[1+(2m)/(M)-1]`
`=(1)/(2)momega_(0)^(2)R^(2)(1+(2m)/(M))`