Question

A spool with thread wound on it, of mass m, rests on a rough horizontal surface. Its moment of inertia relative to its own axis is equal to I = γmR², where γ 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 a 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 F during the first t seconds after the beginning of motion.

Answer 1

(a) Let us choose the positive direction of the rotation angle φ, such that w_cx and β_z have identical signs (Fig.). Equation of motion

Note that at cos α = r/R, there is no rolling and for cosα<r/R, wcx<0, i.e. the spool will move towards negative x-axis and rotate in anticlockwise sense.