Energy in Rotation

Rotational kinetic energy in rigid bodies and mechanical energy conservation.

General idea of the topic

A spinning body stores rotational kinetic energy, ½Iω², analogous to ½mv². The work done by a moment when it turns a body, τ·Δθ, equals the change in that energy, just as the work of a force changes the translational kinetic energy.

The topic develops both results and joins them in rolling, where the total kinetic energy of a body rolling without slipping is the sum of the translational energy of the center of mass and the rotational energy about it.

What you will truly understand

By the end you will be able to compute rotational kinetic energy, the work of a moment and the speed of a rolling body after descending a height, using conservation of energy.

You will also know why a sphere reaches the foot of an incline before a hoop.

Recommended learning route

1. Energy in Rolling: how kinetic energy is shared between translation and rotation in a rolling body.
2. Rotational Kinetic Energy: definition of rotational energy and its dependence on I and ω.
3. Work and Torque: work of a moment and the work-energy theorem for rotation.

Physical criterion you must acquire

The essential criterion is not to forget rotational energy in rolling problems: since energy is shared between translation and spin, the final speed is less than that of frictionless sliding. With rolling without slipping static friction does not dissipate, and mechanical energy is conserved.

Connection with real problems

• flywheels as energy storage.
• vehicles and regenerative braking.
• turbines and generators.
• races of objects rolling down ramps.

Final goal of the topic

The final goal is to apply the energy balance to systems with rotation and compute speeds and heights without solving the equations of motion.