MotionLab

Rotational Kinetic Energy Calculator

The spinning counterpart to ½mv² — rotational KE depends on moment of inertia and angular velocity.

Moment of inertia I (kg·m²)

Angular velocity ω (rad/s)

Rotational KE (J)

KE = ½Iω²

Speed (RPM): 95.493

What you entered

Moment of inertia I (kg·m²): 0.5Angular velocity ω (rad/s): 10

1
Angular velocity in RPM
10 × 60 / (2π)= 95.493 RPM
2
Rotational KE = ½Iω²
0.5 × 0.5 × 10²= 25 J

Result

Rotational KE (J): 25

A body with I = 0.5 kg·m² spinning at 10 rad/s (95.5 RPM) has 25 J of rotational kinetic energy.

Rotational kinetic energy from moment of inertia and spin rate

Rotational energy is half the moment of inertia times the angular velocity squared - the rotational twin of the linear formula.

Moment of inertia (kg m2)	Angular velocity (rad/s)	Energy (J)	Equivalent (rpm)
10	2	20.000	19.099
2	5	25.000	47.746
1	10	50.000	95.493
0.5	20	100.000	190.986
0.1	100	500.000	954.930

Reading down the table, the moment of inertia falls by a factor of 100 while the energy rises tenfold, because angular velocity is squared and inertia is not. That is precisely why flywheels store energy by spinning fast rather than by being heavy, and why a small increase in rpm demands a large increase in stored energy. The moment of inertia itself depends on how mass is distributed relative to the axis, not just how much there is: a hoop and a disc of equal mass have very different values.

Rotational vs. translational kinetic energy

Just as a moving object has KE = ½mv², a spinning object has rotational KE = ½Iω². The moment of inertia I plays the role of mass (how hard it is to spin up), and angular velocity ω replaces linear velocity. A rolling object has both types simultaneously.

Why flywheels store energy

Flywheels exploit ½Iω² for energy storage — by spinning a heavy disk at high speed, significant energy is stored mechanically. Modern flywheel batteries use carbon-fiber rotors spinning at 60,000+ RPM in a vacuum, storing enough energy to power a house for hours.

Frequently asked questions

A bicycle wheel (I = 0.1 kg·m²) spins at 200 RPM. How much rotational KE does it have?

ω = 200 × 2π/60 = 20.94 rad/s. KE_rot = ½Iω² = ½ × 0.1 × 438.6 = 21.9 J. The bike also has translational KE (½mv²) — for a rolling wheel, total KE = translational + rotational, which is why rolling objects are harder to stop than sliding ones.

A flywheel stores 10 MJ of energy at 30,000 RPM. What's its moment of inertia?

ω = 30000 × 2π/60 = 3141.6 rad/s. I = 2E/ω² = 2 × 10⁷ / (3141.6)² = 2.03 kg·m². That's a compact, heavy disk — modern flywheel batteries use carbon-fiber rotors to maximize I while keeping mass manageable.

How is rotational KE different from regular (translational) KE?

Translational KE = ½mv² (straight-line motion). Rotational KE = ½Iω² (spinning). A rolling ball has both — it's moving forward AND spinning. Total KE = ½mv² + ½Iω². For a solid sphere, rotational KE adds 40% on top of translational KE. See the kinetic energy calculator.