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).