Concept #1: Conservation of Energy with Rotation

Example #1: Work to accelerate cylinder

Practice: How much work is needed to stop a hollow sphere of mass 2 kg and radius 3 m that spins at 40 rad/s around an axis through its center?

Example #2: Which shape reaches bottom first?

Example #3: Cylinders racing down: rolling vs. sliding

Practice: Two solid cylinders of same mass and radius roll on a horizontal surface just before going up an inclined plane. Cylinder A rolls without slipping, but cylinder B moves along a slippery path, so it moves without rotating at all times. At the bottom of the incline, both have the same speed at their center of mass. Which will go higher on the inclined plane? (Why?)

A uniform rod of mass m and length l is pivoted about a horizontal, frictionless pin at the end of a thin extension (of negligible mass) a distance l from the center of mass of the rod. The rod is released from rest at an angle of θ with the horizontal, as shown in the figure.
What is the magnitude of the Horizontal force Fx exerted on the pivot end of the rod extension at the instant the rod is in a horizontal position? The acceleration due to gravity is g and the moment of inertia of the rod about its center of mass is 1/12 mℓ2.
1. Fx = 1/13 mg sin(θ)
2. Fx = 24/13 mg cos(θ)
3. Fx = 24/13 mg sin(θ)
4. Fx = 13/12 mg cos(θ)
5. Fx = 12/13 mg cos(θ)
6. Fx = 12/13 mg sin(θ)
7. Fx = 13/12 mg sin(θ)
8. Fx = mg cos(θ)
9. Fx = 1/13 mg cos(θ)
10. Fx = mg sin(θ)

A thin-walled hollow sphere with mass 5.0 kg and radius 0.20 m is rolling without slipping at the base of an incline that slopes upward at 37° above the horizontal. At the base of the incline the translational speed of the center of mass of the sphere is v = 12.0 m/s. If the sphere rolls without slipping as it travels up the incline, what is the maximum vertical height that it reaches before it starts to roll back down?

A solid disk is released from rest and rolls without slipping down an inclined plane that makes an angle of 25.0° with the horizontal. What is the speed of the disk after it has rolled 3.00 m, measured along the inclined plane?
A) 4.07 m/s
B) 6.29 m/s
C) 3.53 m/s
D) 5.71 m/s
E) 2.04 m/s

A thin-walled hollow cylinder (I = MR2), with mass M = 3.00 kg and radius R = 0.200 m, is rolling without slipping at the bottom of a hill. At the bottom of the hill the center of mass of the cylinder has translational velocity 16.0 m/s. The cylinder then rolls without slipping to the top of a hill. The top of the hill is a vertical height of 6.00 m above the bottom of the hill. What is the translational velocity of the center of mass of the cylinder when the cylinder reaches the top of the hill?

A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]

A solid cylinder with moment of inertia I = 1/2 MR2, a hollow cylinder with moment of inertia I = MR2 and a solid sphere with moment of inertia I = 2/5 MR2 all have a uniform density, the same mass and the same radius. They are placed at the top of an inclined plane and allowed to roll down the inclined plane without slipping. Rank them in order of total kinetic energy at the bottom of the incline, from higher to lower.[a] sphere(1st), hollow cylinder(2nd), solid cylinder(3rd)[b] sphere(1st), solid cylinder(2nd), hollow cylinder(3rd)[c] hollow cylinder(1st), sphere(2nd), solid cylinder(3rd)[d] hollow cylinder(1st), solid cylinder(2nd), sphere(3rd)[e] they all have the same kinetic energy

A 5 kg mass hangs from a rope wrapped around the surface of a cylindrical pulley. The pulley has a mass of 20 kg and a radius of 0.8 m. If the 5 kg mass is suddenly released, falling down, what speed will it have after it has fallen 3.2 m?

A thin rod of length 2.8 m and mass 4.2 kg is held vertically with its lower end resting on a frictionless horizontal surface. The rod is then let go to fall freely. Determine the speed of its center of mass just before it hits horizontal surface. The acceleration due to gravity is 9.8 m/s2.
1. 4.53652
2. 3.42929
3. 4.84974
4. 4.92494
5. 4.02119
6. 4.3715
7. 5.07198
8. 4.2
9. 4.77336
10. 3.83406

A uniform hollow disk has two pieces of thin light wire wrapped around its outer rim and is supported from the ceiling (See the figure below). Suddenly one of the wires breaks, and the remaining wire does not slip as the disk rolls down. Use energy conservation to find the speed of the center of this disk after it has fallen a distance of 2.20 m.

A 2 kg mass hangs from a rope wrapped around the surface of a cylindrical pulley. The pulley has a mass of 12 kg and a radius of 0.4 m. If the 2 kg mass is suddenly released, falling down, what angular speed will the cylinder have after the mass has fallen 1.7 m?