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=gammamR^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 (figure). 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.

Explanation

(a) Let us choose the positive direction of the rotation angle `varphi`, such that `w_(cx)` and `beta_z` have identical signs (Fig.) Equation of motion, `F_x=mw_(cx)` and `N_(cz)=I_cbeta_z` gives:
`Fcosalpha-f r=mw_(cx)` `: f r R-F r=I_cbeta_z=gammamR^2beta_z`
In the absence of the slipping of the spool `w_(cx)=beta_zR`
From the three equations `w_(cx)=w_c=(F[cosalpha-(r//R)])/(m(1+gamma))` , where `cos alphagtr/R` (1)
(b) As static friction `(f r)` does not work on the spool, from the equation of the increment of mechanical energy `A_(ext)=Deltat`.
`A_(ext)=1/2mv_c^2+1/2gammamR^2(v_c^2)/(R^2)=1/2m(1+gamma)v_c^2`
`=1/2m(1+gamma)2w_cx=1/2m(1+gamma)2w_c(1/2w_ct^2)`
`=(F^2(cosalpha-r/R)^2t^2)/(2m(1+gamma))`
Note that at `cos alpha=r//R`, there is no rolling and for `cos alphaltr//R`, `w_(cx)lt0`, i.e. the spool will move towards negative x-axis and rotate in anticlockwise sense.