Just like a moving object carries translational kinetic energy, a rotating rigid system carries rotational kinetic energy. It's related to the system's rotational inertia and angular velocity by:
Krot = ½Iω²
🔑Notice the shape of this equation: it's the exact same structure as K = ½mv², with mass swapped for rotational inertia and speed swapped for angular speed. If you already trust ½mv², you already understand the logic behind ½Iω².
And like all kinetic energy, rotational KE is a scalar — it has a magnitude but no direction. It doesn't matter whether the system spins clockwise or counterclockwise; the energy is the same either way.
Adjust the rotational inertia and angular velocity of a spinning rigid body and watch its rotational kinetic energy respond.
I (kg·m²)4
ω (rad/s)6
Krot = ½(4)(6)² = 72.0 J
6.1.A.1.i6.1.A.1.iiConcept
The Total Kinetic Energy of a Rigid System
For an object rotating about a fixed axis — like a wheel spinning on a stationary axle — its rotational kinetic energy about that axis is its total kinetic energy. Every bit of the object's motion is captured by the ½Iω² term.
But plenty of rigid systems both move and spin at once — a rolling ball, a thrown football tumbling through the air, the flywheels and rolling disk in this unit's header art. For those systems, the total kinetic energy splits cleanly into two pieces:
Ktotal = Ktrans + Krot
💡Translational KE comes from the linear motion of the system's center of mass: Ktrans = ½Mv²cm. Rotational KE comes from the system's rotation about that center of mass: Krot = ½Icmω². Add them together and you have the system's complete kinetic energy — nothing left out, nothing double-counted.
Explore the split directly below. Notice that turning either slider to zero doesn't zero out the total — it just means one form of kinetic energy is doing all the work.
A rigid system can move and spin at the same time — like the rolling wheel in this unit's header art. Set its mass, center-of-mass speed, rotational inertia, and angular velocity, and watch how the two kinds of kinetic energy stack up.
ExampleGuided Example — Splitting Total Kinetic Energy
A solid cylinder of mass 4 kg has a center-of-mass speed of 3 m/s and a rotational inertia about its center of mass of 1.5 kg·m², spinning at 8 rad/s. Find its total kinetic energy.
Step 1 — Identify the two pieces
This system is both translating (center of mass moving at 3 m/s) and rotating (spinning at 8 rad/s about that center of mass) — both terms apply.
6.1.A.2Concept⚠ Watch Out
Rotating in Place
Here's a case worth sitting with: a rigid system can have rotational kinetic energy even while its center of mass is completely at rest. A ceiling fan bolted to the ceiling, a record spinning on a turntable, a gyroscope spinning in place on a stand — none of these are going anywhere, yet all of them clearly have kinetic energy. Where does it come from?
🔑It comes from every individual point in the system except the center of mass itself. Each point at some distance r from the rotation axis has its own linear speed v = ωr, and therefore its own kinetic energy. Sum up the contributions from every point in the rigid body, and the result is exactly Krot = ½Iω² — even though the center of mass, sitting right on the axis, has v = 0 the entire time.
⚠️Don't fall into the trap of thinking "center of mass isn't moving, so there's no kinetic energy." That reasoning only checks the translational piece. A system fixed in place can still have all of its kinetic energy tied up in rotation.
ExampleGuided Example — Does It Have Kinetic Energy?
A ceiling fan's center of mass sits exactly on its mounting point and never moves. The blades spin at a constant angular velocity. Does the fan have kinetic energy? Justify your answer.
Step 1 — Check translational KE
The center of mass doesn't move, so vcm = 0, which makes Ktrans = ½Mv²cm = 0.
← Back to Lesson 6.1Ready for 6.2? Torque and Work looks at how a torque acting through an angular displacement transfers energy into or out of a rotating system.