A physics teacher uses a ramp making an angle of 20.° with the horizontal to show the students the rolling motion of different objects. First, she releases from rest a disk of diameter d from a height of 50. cm above the floor. Then, she releases a uniform spherical ball of diameter d from a height of h. The two objects reach the end of the ramp at the same speed. Find h. Consider that the two objects roll to the bottom without slipping.

A cylindrical cell is given an initial speed of v₀ at the bottom of a smooth ramp that makes an angle of 20° with the horizontal. The cell rolls up 1.3 m without slipping before stopping briefly and rolling down. The roughness of the ramp is increased, and the cell is given the same initial speed of v₀ at the bottom of the ramp. The cell rolls up without slipping a distance of 1.1 m. Calculate the coefficient of rolling friction of the rough ramp.

Suppose a uniform solid cylinder reaches the bottom of a slope 7.21 m high and rolls without slipping. Calculate the cylinder's uniform translational speed. Assume the cylinder starts from rest at the top of the slope.

A narrow, hollow 0.546-kg tube section with a radius of 13.0 cm rolls (from a standstill) down a 17.6° ramp of 5.61 m in length. Calculate the ramp's base total kinetic energy.

A toy car shaped like a sphere with mass m and radius r is placed on a toy track that includes a vertical loop of radius R. Find the minimum height h from which the toy must start to complete the loop if (A) r<<R, (B) no assumption of r with R, assuming it rolls without slipping and there are no energy losses due to friction or air resistance.

A composite object is made up of two solid cylindrical disks connected by a light rod of negligible mass, and it starts from a position of rest. The smaller disk has a mass of 0.025 kg and a radius of 0.05 m, while the larger disk has a mass of 0.15 kg and a radius of 0.10 m. This setup is arranged to rotate around the common axis of the cylinders and is dropped from a height of 2.0 meters without any initial rotation, in a situation where air resistance is ignored.

(a) Use the principle of conservation of energy to calculate the linear speed of the system just before it hits the ground.

(b) What fraction of its kinetic energy would be rotational just before impact?

A hollow, slender 0.546-kg ring with a radius of 13.0 cm rolls (from a standstill) down a 17.6° slope of 5.61 m in length. What is the minimum static friction coefficient if the ring does not slip?

During a movie shoot, a car with a stunt driver inside is lifted off the ground. The stunt driver starts the engine and sets the wheels of the car with angular speeds of ω_i. The car then is brought back down onto the ground. At the moment the car wheels make contact with the ground, the speeds of the center of masses of the car wheels are zero. In the beginning, the car wheels skid on the ground and move forward. But after a short time, they begin to roll without skidding. Given that, the moment of inertia of a car wheel about the axle is I = (1/2)mR², and the coefficient of friction between the car wheels and the ground is μ, evaluate what the final translational speed value v_cm of the center of mass of a car wheel will be.

[Hint: Use Newton's second law, ∑τ_cm = I_cmα_cm, and consider that v_cm = ωr only applies when the car wheels roll without skidding.]