Comparison of Total Kinetic Energy for Rolling Hoops on Incline Planes

Which hoop will have the greater total kinetic energy at the bottom?

Answer:

They both have the same total kinetic energy at the bottom.

Explanation:

When two hoops, starting from rest, roll down identical incline planes with the work done by nonconservative forces being zero, the total kinetic energy at the bottom for both hoops will be the same.

This is due to the conservation of total mechanical energy, which means that the initial total kinetic energy and potential energy of the hoops must equal the final total kinetic energy and potential energy. If both hoops start from rest and at the bottom of the incline the level for gravitational potential energy is zero for reference, then the kinetic energy at the bottom can be calculated as:

K1 = 0, U2 = 0. Therefore, ΔK = ΔU = mgh.

Given that both inclines have the same height and both hoops have the same mass m, the difference in kinetic energy between the two hoops must be equal, resulting in them having the same total kinetic energy at the bottom.