A uniform hoop of mass m and radius r rolls without slipping on a fixed cylinder of radius R. The only external force is that of gravity. If the smaller cylinder starts rolling from rest on top of the bigger cylinder, use the method of Lagrange multipliers to find the point at which the hoop falls off the cylinder.

A bicycle wheel is spinning at a constantspeed. You note that a piece of tape on theedge of the wheel completes 25 fullrotations in 19 seconds. What is the angularfrequency of the bicycle wheel?

In the figure below, the disk, of mass 440 g and radius 3.5 cm, is rotating at 180 rpm on a frictionless shaft of negligible radius. The upper disk, of mass 270 g and radius 2.3 cm, is initially not rotating. It drops freely down onto the lower dis, and frictional forces bring the two disk to a common rotational speed. Find (a) that common speed and (b) the fraction of the initial kinetic energy lost to friction.

The uniform thin rod in the figure below has mass M = 2.00 kg and length L = 3.25 m and is free to rotate on a frictionless pin. At the instant the rod is released from rest in the horizontal position, find the magnitude of the rod's angular acceleration, the tangential acceleration of the rod's center of mass, and the tangential acceleration of the rod's free end.

A uniform, solid sphere of radius 6.00 cm and mass 4.50 kg starts with a purely translational speed of 2.50 m/s at the top of an inclined plane. The surface of the incline is 2.00 m long, and is tilted at an angle of 32.0∘ with respect to the horizontal. Assuming the sphere rolls without slipping down the incline, calculate the sphere's final translational speed v2 at the bottom of the ramp. v2=   m/s

A frisbee’s stability is maintained by its angular momentum. A 175gram frisbee with a diameter of 12inchesis thrown with an angular velocity of 5rev/s . Assume the frisbee can be modeled as a flat solid disk. Find the magnitude of the angular momentum vector.

Asmall cube (M = 0.45 kg) is at a height of 315 cm up a frictionless track which has a loop of radius, R= 47.25 cm at the bottom. The %3D cube slides freely down the ramp and around the loop. Find the velocity of the block when it is at the top of the loop. V H R