1) For practice: Prove the center of mass of a solid uniform square plate of mass M and side length S will be at its center using calculus. Treat dm as a strip with infinitesimal width and length S.

A person of mass mp is standing at the right end of a cart of mass me and length s. The cart and the person are initially at rest. The person then walks to the left end and stops. You may assume that there is zero rolling resistance between the cart and the ground. (Part a) What is the speed of the cart when the person has finished walking? Express your answer only in terms of mp, mc and s as needed. Vc, f = (Part b) When the person is finished walking in what direction and how far has the cart moved? Express your answer only in terms of mp, me and s as needed. Xc₂f = (Part c) If the person takes a time interval At to go from rest to a speed u relative to the cart, what is the average force the person exerts on the cart during this time interval? Express your answer only in terms of mp, mc, Deltat for At and u as needed. Fp,c ▸

Equilibrium II A massless wheel of radius R is mounted on a frictionless, horizontal axle. A point mass M is glued to the edge of the wheel, and a mass m hangs from a string wrapped around the perimeter of the wheel. (a) Write down the total PE of the two masses as a function of the angle O. (b) Use this to find the values of m and M for which there are any positions of equilib- rium. Describe the equilibrium positions and their stability. (c) Find the critical value of m/M other side of which it does not (if released from rest at o= 0). on one side of which the system oscillates and on the R М m

A thin disk of initial mass M and radius R has a hole of radius r removed. The center of the hole is a distance r away from the center of the disk. Disk R r. (a) Before doing any calculations, mark on the figure the appoximate position of the center of mass of the object now that it has this hole. (b) Find the exact position of the center of mass of the object.