The kinetic energy of a spinning wheel is E = ½ × I × ω², where I is the moment of inertia in kg·m² and ω is the angular speed in radians per second (rpm × 2π ÷ 60). For a thin ring I = m × r² and for a solid disc I = ½ × m × r², so a 500 kg ring of 0.6 m radius at 1,500 rpm stores 2,221 kJ, and the same mass as a solid disc stores half that.
Part of Flywheel Energy Storage
Where the flywheel energy formula comes from, how to evaluate it in the right units, and how to check any quoted flywheel figure.
The Formula Is Just Added-Up Moving Mass
A rotating wheel is a very large number of small pieces, each of mass mᵢ at distance rᵢ from the axis. Every piece moves at speed vᵢ = ω × rᵢ. Add up the ordinary kinetic energy of all of them:
E = Σ ½ × mᵢ × vᵢ² = ½ × (Σ mᵢ × rᵢ²) × ω² = ½ × I × ω²
The bracket is the moment of inertia I. It is the mass, weighted by the square of its distance from the axis, so a kilogram at the rim counts far more than a kilogram near the hub.
Moment Of Inertia For The Two Shapes That Matter
Thin ring (all mass at one radius r): every piece has rᵢ = r, so I = m × r².
Solid disc of radius R, thickness t, density ρ: slice it into rings of radius r and width dr. Each ring has mass dm = ρ × t × 2π × r × dr, and contributes r² × dm to I.
I = ∫ from 0 to R of ρ × t × 2π × r³ dr = ρ × t × 2π × R⁴ ÷ 4
The disc’s mass is m = ρ × t × π × R², so I = ½ × m × R². The disc’s I is half the ring’s for the same mass and radius. The practical consequences, including a result that runs against the usual teaching once strength is considered, are in Rim Mass vs Solid Disc.
Worked Example: Check A Quoted Table
The third row below is a 500 kg ring of 0.6 m radius at 1,500 rpm, the wheel used throughout this series.
The carrier page Flywheel Energy Storage lists four wheels. Recomputed here from its own masses, radii and speeds (ring or disc as noted):
| Wheel | m, r, speed | I (kg·m²) | ω (rad/s) | E |
|---|---|---|---|---|
| Cast iron disc | 100 kg, 0.25 m, 1,500 rpm | ½ × 100 × 0.0625 = 3.125 | 157.1 | ½ × 3.125 × 157.1² = 38.6 kJ (10.7 Wh) |
| Steel ring | 150 kg, 0.40 m, 2,000 rpm | 150 × 0.16 = 24 | 209.4 | ½ × 24 × 209.4² = 526 kJ (146 Wh) |
| Heavy steel ring | 500 kg, 0.60 m, 1,500 rpm | 500 × 0.36 = 180 | 157.1 | 2,221 kJ (617 Wh) |
| Large steel ring | 2,000 kg, 1.00 m, 1,000 rpm | 2,000 × 1 = 2,000 | 104.7 | ½ × 2,000 × 104.7² = 10,966 kJ (3,046 Wh) |
Those results are 2.0, 4.0, 6.7 and 10 times larger than the carrier’s printed 19, 132, 333 and 1,097 kJ. The error matters because every one of those rims is under stress proportional to speed squared, and a reader who believes the smaller energy will believe the wheel is less dangerous than it is.
Units Decide The Answer
- rpm to rad/s: ω = rpm × 2π ÷ 60 = rpm × 0.10472. 1,500 rpm is 157.1 rad/s.
- Mass, not weight: m is in kilograms. A “500 kg” wheel is 4,905 N; putting 4,905 in for m overstates the energy 9.81 times.
- Joules to watt-hours: divide by 3,600.
- Tip speed: v = ω × r. It is the number that later sets the stress in the rim.
Power, Not Just Energy
Torque on the shaft relates to the rate at which the wheel slows: T = I × (dω ÷ dt). A 180 kg·m² wheel losing speed at 1 rad/s² delivers 180 N·m, and at 157 rad/s that is 28 kW. The wheel can supply large power briefly, but only until the energy is gone: 2,221 kJ at a constant 28 kW would last 79 s.
Common Mistakes
- Using the disc formula for a ring or the reverse. It is a factor of 2.
- Using diameter for r. Radius enters squared; a diameter error is a factor of 4.
- Forgetting that the formula says nothing about whether the wheel survives. See Burst Energy Calculation.
Safety
Do not trust a printed energy figure
A flywheel table that understates the energy by 2 to 10 times understates the hazard by the same factor. Recompute from mass, radius and speed before you decide where the wheel can sit or what must stand between it and people. 2.2 MJ from a half-tonne wheel is already enough to throw a fragment hundreds of metres.
What’s Next
- Learn next: Rim Mass vs Solid Disc — which shape stores more, and at what stress
- Combine with: Flywheel Design — sizing a wheel to smooth a machine rather than to bank energy