The uniform slender bar AB has mass mAB and length L. The crate has a uniformly distribute...

The uniform slender bar AB has mass mAB and length L. The crate has a uniformly distributed mass mC and dimensions h and w. Bar AB is pin-connected to the trolley at A and to the crate at B. The trolley is constrained to move along the horizontal guide shown. Point O on the trolley’s guide is a fixed reference point.

Figure P7.84

Neglecting the mass of the trolley and friction, derive the equations of motion of the system and express them in terms of the variables xA, θ, and ϕ along with their time derivatives.