Rotational kinetic energy is the energy an object has because it is spinning. It matters whenever wheels, gears, flywheels, planets, disks, or rolling balls are in motion. A spinning object can store energy even if its center is not moving from place to place.
The amount of energy depends on both how fast it rotates and how its mass is distributed around the axis.
Key Facts
- Rotational kinetic energy: Krot = 1/2 I omega^2
- Translational kinetic energy: Ktrans = 1/2 m v^2
- Moment of inertia measures rotational resistance: larger I means harder to spin up or slow down.
- For rolling without slipping: v = omega R
- Total kinetic energy of a rolling object: Ktotal = 1/2 m v^2 + 1/2 I omega^2
- Common moments of inertia: solid cylinder I = 1/2 m R^2, hoop I = m R^2, solid sphere I = 2/5 m R^2
Vocabulary
- Rotational kinetic energy
- The energy an object has due to spinning about an axis.
- Moment of inertia
- A measure of how strongly an object's mass distribution resists changes in rotational motion.
- Angular velocity
- The rate at which an object rotates, usually measured in radians per second.
- Rolling without slipping
- Motion where the point of contact with the ground is momentarily at rest and v = omega R.
- Axis of rotation
- The line around which an object spins.
Common Mistakes to Avoid
- Using mass instead of moment of inertia in Krot = 1/2 I omega^2. Rotational kinetic energy depends on how mass is distributed, not just on total mass.
- Forgetting to include translational kinetic energy for rolling objects. A rolling wheel usually has both Ktrans and Krot, so the total is not just one term.
- Using degrees per second for omega in energy formulas. The formula Krot = 1/2 I omega^2 requires angular velocity in radians per second.
- Assuming all rolling objects with the same mass and speed have the same kinetic energy. Different shapes have different moments of inertia, so they store different amounts of rotational energy.
Practice Questions
- 1 A solid cylinder has mass 4.0 kg, radius 0.30 m, and angular velocity 12 rad/s. Using I = 1/2 m R^2, find its rotational kinetic energy.
- 2 A hoop of mass 2.5 kg and radius 0.40 m rolls without slipping at 3.0 m/s. Find its total kinetic energy using I = m R^2 and v = omega R.
- 3 A solid sphere and a hoop have the same mass, radius, and center-of-mass speed while rolling without slipping. Which has more total kinetic energy, and why?