Problem 1.254

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 \(\mathbf{w}_{0}\). Find the acceleration \(\mathbf{w}^{\prime}\) of the cylinder relative to the car and the force \(F\) exerted by the cylinder on the ceiling (through the threads).