A string is attached to the drum (radius r) of a spool (radius R) as shown in side and end views here. (A spool is device for storing string, thread etc.) A tension T is applied to the string at angle ! above the horizontal. The coefficients of kinetic and static friction between floor and spool are µk and µs respectively. We are interested in whether and when the spool will move left or right, and how this depends on the nature of the floor.
(a) Draw a free body diagram for the spool, showing the forces acting on it, for the case when it is in mechanical equilibrium. (The force vectors need not be to scale, but they should be in approximately the correct direction.)
(b) If it slides or skids, in which direction will it move when pulled by the string?
(c) If it rolls, in which direction will it move when pulled by the string?
(d) Showing all working, calculate the critical value of ! (call it !c) at which the condition goes from rolling to skidding.