Problem 1.257

A spool with thread wound on it, of $\mathrm{mass} m,$ rests on a rough horizontal surface. Its moment of inertia relative to its own axis is equal to $I=\gamma m R^{2},$ where $\gamma$ is a numerical factor, and $R$ is the outside radius of the spool. The radius of the wound thread layer is equal to $r$. The spool is pulled without sliding by the thread with a constant force $F$ directed at an angle $\alpha$ to the horizontal (Fig. 1.63). Find: (a) the projection of the acceleration vector of the spool axis on the $x$ -axis; (b) the work performed by the force $\mathbf{F}$ during the first $t$ seconds after the beginning of motion.