Thin threads are tightly wound on the ends of a uniform solid cylinder of mass m. The free ends of the threads are attached to the ceiling of an elevator car. The car starts going up with an acceleration `w_0`. Find the acceleration `w^'` . Find the acceleration `w^'` of the cylinder relative to the car and the force F exerted by the cylinder on the ceiling (through the threads).

Explanation

Let us depict the forces and their points of application corresponding to the cylinder attached with the elevator. Newton's second law for solid in vector form in the frame of elevator, gives:
`2vecT+mvecg+m(-vecw_0)=mvecw` (1)
The equation of moment in the C.M. frame relative to the cylinder axis i.e. from `N_z=I_cbeta_z`.
`2TR=(mR^2)/(2)beta=(mR^2)/(2)(w^')/(R)`
[as thread does not slip on the cylinder, `w^'=betaR`]

or, `T=(mw^')/(4)`
As (1) `vecTuarrdarrvecw`
so in vector form
`vecT=-(mvecw)/(4)` (2)
Solving Eqs. (1) and (2), `overset(rarr')w=2/3(vecg-vecw_0)` and sought force
`vecF=2vecT=1/3m(vecg-vecw_0)`