Hong Life CalculatorsUnderstanding the flywheel energy storage calculator — concept and principle
This calculator finds how much kinetic energy a spinning flywheel stores and how much of it you can actually use between a maximum and a minimum working speed. It also gives the moment of inertia, rim speed and specific energy, which are useful for engine flywheels, uninterruptible power supplies, regenerative systems and physics homework.
How it is calculated
E = ½ I ω², with ω = rpm × 2π ÷ 60
Usable energy = ½ I (ω_max² − ω_min²)
Moment of inertia: solid disc ½ m r²; thin rim m r²; thick ring ½ m (r_i² + r_o²)
Rim speed = ω × r_o; specific energy = E ÷ m
- Mass in kg, radii in m, speeds in rpm; energy in kJ and Wh (1 Wh = 3,600 J).
The principle
A rotating body stores energy in the motion of every bit of its mass, and mass further from the axis moves faster, so it counts more. The moment of inertia sums mass times distance squared, which is why a rim stores twice as much as a solid disc of the same mass and radius. Energy grows with the square of speed, so the top of the speed range holds most of it: dropping from full to half speed releases 75% of the stored energy.
Worked example
Default inputs: solid disc, 100 kg, r_o = 0.3 m, 10,000 rpm maximum, 5,000 rpm minimum.
- I = ½ × 100 × 0.3² = 4.5 kg·m².
- ω_max = 10,000 × 2π ÷ 60 = 1,047.2 rad/s.
- E = ½ × 4.5 × 1,047.2² = 2,467.4 kJ = 685.4 Wh.
- Usable = 685.4 × (1 − 0.5²) = 514 Wh, or 75%.
- Rim speed = 1,047.2 × 0.3 = 314 m/s; specific energy = 685.4 ÷ 100 = 6.85 Wh/kg.
| Outer radius r_o (m) | Usable energy between the speeds (Wh) |
|---|---|
| 0.075 | 32.1 |
| 0.15 | 128.5 |
| 0.225 | 289.1 |
| 0.3 | 514 |
| 0.375 | 803.2 |
| 0.45 | 1,156.6 |
| 0.6 | 2,056.2 |
How to read the result
The usable energy, not the total, is what a flywheel can deliver before speed drops below what the application allows. Rim speed is the real limit: the stress in the rotor rises with its square, so the safe speed depends on material and construction and must come from the manufacturer. Steel rotors have modest specific energy; composite rotors allow much higher rim speeds.
Common mistakes
- Using rpm directly in E = ½Iω² instead of rad/s.
- Treating a spoked wheel as a solid disc.
- Counting all the stored energy as usable.
- Ignoring the rim-speed limit when increasing speed.
Frequently asked questions
Why does doubling speed quadruple energy?
Kinetic energy depends on ω², so twice the speed means four times the energy.
Does a rim store more energy than a disc?
Yes, for the same mass and outer radius, twice as much, because all its mass is at the largest radius.
Are bearing losses included?
No. Real flywheels lose energy to bearings and air drag over time.
Sources: OpenStax, University Physics Volume 1 (2016), section 10.4 "Moment of Inertia and Rotational Kinetic Energy"