Rotational kinetic energy is the kinetic energy associated with a rigid system spinning about an axis. It is a scalar — always positive or zero, regardless of the direction of rotation.
The existence of rotational KE follows directly from the fact that every point on a rotating system has a linear speed v = rω. Because moving mass has kinetic energy (½mv²), every small piece of the rotating object carries energy. The sum of all those contributions is the rotational kinetic energy of the whole system.
For a rigid system with rotational inertia I rotating at angular velocity ω:
This mirrors the translational formula exactly: I replaces m, ω replaces v. The rotational inertia I captures how much mass there is and where it sits relative to the axis. The ω² factor means angular velocity has a quadratic effect — doubling ω quadruples K_rot, just as doubling v quadruples ½mv².
A solid disk (mass 4 kg, radius 0.5 m) spins at 6 rad/s about its central axis. Calculate its rotational kinetic energy.
A gyroscope spinning in a fixed mount has zero translational kinetic energy — its center of mass is stationary. Yet it carries substantial rotational KE, because every atom in the gyroscope is moving in a circle.
This is counterintuitive but follows directly from v = rω. A point on the rim at radius r has linear speed v = rω even when the center of mass never moves. That linear speed means kinetic energy. Sum it over all the mass in the object and you get K_rot = ½Iω².
A flywheel (I = 2 kg·m²) spins at 10 rad/s with its axle fixed. A student claims it has zero kinetic energy because it is not going anywhere. Who is right and what is the actual KE?
For an object that both translates and rotates — like a ball rolling across a floor — the total kinetic energy is the sum of both contributions:
The first term captures the energy of the center of mass moving through space. The second captures the energy of rotation about the center of mass. These are independent energy accounts — both real, both contributing to the total.
Adjust rotational inertia, angular velocity, mass, and linear speed. Watch how rotational and translational KE combine into total KE. A spinning object with a moving CM carries both simultaneously.
Set v_cm = 0 — the object spins in place. K_trans = 0 but K_rot is real and nonzero. Set omega = 0 — pure translation. Set both — a rolling object carrying both energy accounts.
A solid sphere (mass 2 kg, radius 0.1 m, I = 2/5 MR²) rolls without slipping at v_cm = 5 m/s. Find its total kinetic energy.
The work-energy theorem from Unit 3 extends directly to rotation. The net work done on a rotating system by all torques equals the change in rotational kinetic energy:
Where W_net is the total work done by all torques (W = τΔθ from Lesson 6.2). A net torque in the direction of rotation does positive work and increases K_rot. A torque opposing rotation does negative work and decreases K_rot — this is how friction slows a spinning object.
A 3 kg mass falls 0.8 m and drives a solid disk pulley (mass 2 kg, radius 0.2 m, I = ½MR²). Starting from rest, find the final speed of the mass. Assume no friction.