About the Rotational Kinetic Energy Calculator
This rotational kinetic energy calculator finds the energy stored in a spinning object using KE = ½Iω². Choose a common shape — solid disk or cylinder, thin ring, solid or hollow sphere, or a rod — and enter its mass, radius or length and spin rate, and it works out the moment of inertia for you. If you already know the moment of inertia, enter it directly.
It suits physics students checking rotational dynamics problems, engineers sizing a flywheel for energy storage, and makers curious how much energy a spinning wheel, grinder disc or rotor carries. Speed can be entered in rpm, rad/s, rev/s or degrees per second, and the result is given in joules and watt-hours, along with the rim speed. A chart shows how the energy climbs with the square of the speed — doubling the rpm stores four times the energy.
For a ball, wheel or cylinder rolling without slipping, switch on “rolling” to add the translational kinetic energy ½mv² (with v = ωr) and see the total. The shape formulas assume uniform density and rotation about the stated axis; real flywheels with a heavy rim sit between the disk and the ring values.
With the default inputs, the rotational kinetic energy is 22,206.61 J. Change any value above to recalculate instantly.
How to use the rotational kinetic energy calculator
- 1Pick the shape of the spinning object, or choose to enter the moment of inertia directly.
- 2Enter the mass and the radius (or rod length).
- 3Enter the spin rate and choose rpm, rad/s, rev/s or °/s.
- 4For a rolling ball, wheel or cylinder, switch on rolling to include its forward motion.
- 5Read the rotational energy in joules and watt-hours, plus the moment of inertia and rim speed.
Formula and method
A rotating body stores kinetic energy because every particle in it moves in a circle. Adding up ½mv² for all particles gives ½Iω², where I, the moment of inertia, measures how far the mass sits from the axis and ω is the angular velocity in radians per second. Speeds in rpm are converted with ω = rpm × 2π/60.
For standard uniform shapes I = k·m·r², with k = ½ for a solid disk or cylinder, 1 for a thin ring, ⅖ for a solid sphere and ⅔ for a thin hollow sphere; a thin rod uses its length L with k = 1/12 about its centre and ⅓ about one end. An object rolling without slipping also moves forward at v = ωr, so its total kinetic energy adds ½mv². Energy in watt-hours is joules divided by 3,600.
- KE_rot
- Rotational kinetic energy (J)
- I
- Moment of inertia (kg·m²)
- ω
- Angular velocity (rad/s)
- m
- Mass (kg)
- r, L
- Radius or rod length (m)
- k
- Shape factor for the moment of inertia
- v
- Rim speed, or forward speed when rolling (m/s)
Worked examples
10 kg flywheel disk spinning at 3,000 rpm
I = ½ × 10 × 0.3² = 0.45 kg·m² and ω = 3,000 × 2π/60 = 314.16 rad/s, so KE = ½ × 0.45 × 314.16² = 22,207 J, or about 6.17 Wh. The rim moves at 0.3 × 314.16 = 94.2 m/s.
Known moment of inertia: 2 kg·m² at 10 rad/s
KE = ½ × 2 × 10² = 100 J.
Rolling 7 kg ball (r = 10 cm) at 8 m/s
Rolling at 8 m/s with r = 0.1 m means ω = 80 rad/s. Rotational KE = ½ × (⅖ × 7 × 0.1²) × 80² = 89.6 J and translational KE = ½ × 7 × 8² = 224 J, for 313.6 J in total — about 29% of it rotational.
1.5 kg rod swung from one end at 2 rev/s
I = ⅓ × 1.5 × 1² = 0.5 kg·m² and ω = 2 × 2π = 12.566 rad/s, so KE = ½ × 0.5 × 12.566² = 39.5 J. The tip moves at 12.57 m/s.
Frequently asked questions
What is the formula for rotational kinetic energy?+
KE = ½Iω², where I is the moment of inertia in kg·m² and ω the angular velocity in rad/s. It is the rotational counterpart of ½mv² for straight-line motion.
How do I convert rpm to rad/s for this formula?+
Multiply rpm by 2π and divide by 60, i.e. by about 0.10472. 3,000 rpm is 314.16 rad/s. Using rpm directly in ½Iω² gives a wrong answer by a factor of about 91.
How much energy can a flywheel store?+
Energy grows with the square of speed and with the moment of inertia, so heavy rims and high rpm store the most. A 10 kg, 30 cm disk at 3,000 rpm holds about 22 kJ (6 Wh); commercial steel or carbon-fibre flywheels spin at tens of thousands of rpm to store kilowatt-hours.
What is the moment of inertia of a disk?+
A uniform solid disk or cylinder spinning about its central axis has I = ½mr². A thin ring of the same mass and radius has I = mr², so it stores twice as much energy at the same speed.
What is the total kinetic energy of a rolling object?+
A rolling object both spins and moves forward, so KE_total = ½Iω² + ½mv² with v = ωr. A solid sphere has 2/7 of its kinetic energy in rotation, a solid cylinder 1/3, and a thin hoop 1/2.
Why does energy depend on the square of speed?+
Each bit of mass moves at a speed proportional to ω, and kinetic energy goes as speed squared. So doubling the rpm quadruples the stored energy, and halving it leaves only a quarter.