Rotational kinetic energy
A rotating object stores kinetic energy in its spin, given by one half its moment of inertia times its angular velocity squared — the rotational counterpart of one half mv².
Practise this conceptA wheel spinning on a fixed axle is going nowhere, and yet stopping it takes real work — so it plainly holds energy. That energy is ½Iω², the rotational twin of ½mv², with moment of inertia standing in for mass and angular velocity for speed. The two expressions are the same statement about motion, written once for travelling and once for turning.
Because of the square, spin rate dominates. Double the angular velocity and the stored energy quadruples; this is exactly why a flywheel is a serious energy store and also why a burst one is dangerous. And because the moment of inertia depends on how the mass is distributed, a flywheel is built heavy at the rim: the same kilogram stores far more energy out there than near the hub.
An object that rolls is doing both things at once, and its kinetic energy is the sum: ½mv² for the centre of mass moving along, plus ½Iω² for the spin about that centre. Miss the second term and rolling problems come out wrong. It is the reason a ball released on a slope arrives slower than a frictionless block released beside it — some of the gravitational energy went into spinning it up rather than into speeding it along.
Key ideas
Requires: Moment of inertia and rotational dynamics, Work and energy
Unlocks: Lagrangian mechanics, Rolling motion
Formulas
| Symbol | Name | Unit |
|---|---|---|
| energy | J (joule) | |
| moment of inertia | kg*m^2 (kilogram square metre) | |
| angular velocity | rad/s (radian per second) |
Common misconceptions
- An object spinning in place has no kinetic energy, because its centre of mass is not going anywhere.
- A rolling object's kinetic energy is just ½mv², the same as if it were sliding.
- Doubling the angular velocity doubles the rotational kinetic energy.
Resources
Part of these UT courses
This concept is mapped as covering part of the following course(s) in the University of Tartu curriculum.
- Physics practical I — mechanics and heatLTFY.01.004
- Mechanics and heatLTFY.01.006
- Analytical mechanicsLTFY.04.016