How to Calculate the Kinetic Energy of a Rolling Hoop Going Up a Ramp

What physics concepts are involved when a large hoop rolls without slipping up a ramp?

1. Angular velocity

2. Translational and rotational kinetic energy

3. Moment of inertia

Answer:

The question deals with physics, specifically the concepts of angular velocity, translational and rotational kinetic energy, and moment of inertia in the context of a hoop rolling without slipping.

Explanation:

In the question, we are dealing with a circular hoop rolling up a ramp. For the hoop to roll without slipping, its angular speed and its linear speed are related by v = rω, where r is the radius of the hoop, and ω is the angular speed.

The kinetic energy of the rolling hoop just as it starts up the ramp is obtained through combining the rotational kinetic energy and translational kinetic energy. Hence, the total kinetic energy of the hoop can be given by K.E = 1/2 m(rω)² + 1/2 Iω², where I is the moment of inertia of the hoop and m is the mass. Using this equation, one can calculate the kinetic energy when the hoop starts to roll up the ramp.

Now, the moment of inertia comes into play. For a thin hoop rotating about an axis perpendicular to the plane of the hoop, the moment of inertia can be given by I = mR². Thus, the kinetic energy equation becomes K.E = 1/2 m(rω)² + 1/2 mR²ω². This is the energy the hoop has to climb up the ramp.