Meriam, Kraige & Bolton — Plane Kinetics of Rigid Bodies: Problem 6_18

6/18 The thin hoop of negligible mass and radius $r$ contains a homogeneous semicylinder of mass $m$ which is rigidly attached to the hoop and positioned such that its diametral face is vertical. The assembly is centered on the top of a cart of mass $M$ which rolls freely on the horizontal surface. If the system is released from rest, what $x$-directed force $P$ must be applied to the cart to keep the hoop and semicylinder stationary with respect to the cart, and what is the resulting acceleration $a$ of the cart? Motion takes place in the $x-y$ plane. Neglect the mass of the cart wheels and any friction in the wheel bearings. What is the requirement on the coefficient of static friction between the hoop and cart?