The flywheel energy calculator above treats a rotor as an energy store rather than as a spinning body. That distinction drives everything on this page: a store is judged by how much energy you can get back out of it, over what speed range, at what power, and per kilogram of rotor — not simply by how much it contains at full speed.
Arb Digital builds free physics calculators that each own one job. The rotational kinetic energy calculator gives one half I omega squared for any spinning body, along with its angular momentum and rolling energy. This page starts from the same expression and takes it into storage territory: a usable window between two working speeds, a depth-of-discharge fraction, specific energy in watt-hours per kilogram, and the rim speed the geometry has to survive.
What This Flywheel Energy Calculator Does
The tool computes the moment of inertia from the rotor mass, its outer radius and an inertia constant that depends on how the mass is distributed. It then evaluates the stored energy at the maximum speed and at the minimum usable speed, and reports the difference. That difference is the number a designer actually specifies against, because a flywheel is never run to a standstill in service.
Alongside it the tool gives specific energy per kilogram of rotor, which is the honest basis for comparing flywheel storage against batteries or compressed air, and the rim speed at the outer radius. Rim speed is the hidden constraint in every flywheel: the stress in a rotating rotor depends on the square of the tip speed and on material density, not on the rotational speed alone, so a large slow wheel and a small fast one can be equally close to bursting.
The discharge-time input turns the usable energy into an average power. Flywheels are almost always specified by power rather than energy — a ride-through system might be asked for 100 kW for fifteen seconds — and the conversion between the two is where a plausible-looking design usually falls apart.
How to Use It
- Pick the geometry. A solid disc has an inertia constant of one half, a thin rim has one, and a hub-mounted thick rim sits between them. If you already have a moment of inertia from CAD or a datasheet, switch to direct entry and use it.
- Enter the rotor mass and outer radius. Use the rotor alone, not the whole machine. Shaft and housing mass contribute nothing to storage and would inflate the specific energy figure.
- Set the maximum and minimum speeds. The minimum is the operational floor below which the load or the power electronics stops working, not zero.
- Add a discharge time. The tool then reports the average power the store can hold over that interval, which is the specification most systems are actually bought against.
- Check the rim speed before anything else. If it is beyond what the rotor material can carry, the energy figure is describing a rotor that would fail, and no other number on the page matters.
The Formula: How Flywheel Energy Is Calculated
Stored rotational energy is E = ½Iω², with I the moment of inertia in kilogram metres squared and ω the angular speed in radians per second. OpenStax University Physics Volume 1, section 10.4 on moment of inertia and rotational kinetic energy, derives this by summing the kinetic energy of every mass element and shows why the moment of inertia takes the place of mass in the rotational form.
The moment of inertia is written I = kmr², where k depends only on how the mass is distributed radially. OpenStax section 10.5 on calculating moments of inertia works the uniform disc case by integration and obtains one half m r squared; a thin hoop, with all its mass at the outer radius, gives m r squared and therefore twice the storage for the same mass and diameter.
Angular speed in radians per second is rpm multiplied by 2π and divided by 60. Rim speed is v = ωr. Usable energy is the difference of the two stored energies, which simplifies neatly to Eusable = ½I(ωmax² − ωmin²), and the recoverable fraction is 1 − (ωmin/ωmax)² — a function of the speed ratio alone, independent of the rotor.
Work the defaults by hand. A solid disc of 100 kg and 0.5 m radius has I = 0.5 × 100 × 0.25 = 12.5 kg·m². At 10,000 rpm, ω = 10,000 × 2π ÷ 60 = 1,047.198 rad/s, so E = 0.5 × 12.5 × 1,047.198² = 6.854 MJ, which is 1.904 kWh. At 5,000 rpm the stored energy is a quarter of that, 1.713 MJ, so the usable energy is 5.140 MJ or 1.428 kWh — exactly 75 per cent, as the speed-ratio formula predicts. Rim speed at full speed is 1,047.198 × 0.5 = 523.6 m/s.
Why Rim Speed, Not RPM, Is the Real Limit
The stress in a spinning rotor is what eventually destroys it, and to a good first approximation the peak stress in a thin rim is the material density multiplied by the square of the rim speed. Density and rim speed are the only two things in that expression. Radius does not appear on its own, and neither does rpm.
The consequence is that the maximum energy a flywheel can store per kilogram depends only on the ratio of the material's tensile strength to its density, multiplied by a shape factor. That is why flywheel development moved to carbon fibre composites: not because they are strong in absolute terms — good steel is stronger — but because their strength-to-density ratio is several times better, and it is the ratio that sets the ceiling.
It also explains why doubling the diameter of a steel rotor buys you far less than it appears to. To keep the same rim speed you must halve the rpm, and the energy stays where it was. The gain from a bigger wheel comes only from the extra mass, not from the geometry. Our hoop stress calculator works the stress side of this problem for a cylindrical wall, and it is the natural companion to the energy figure here.
How Much Energy You Can Actually Get Out
Because energy scales with the square of speed, the recoverable fraction is fixed by the speed ratio and by nothing else. Slowing to 50 per cent of top speed releases 75 per cent of the stored energy. Slowing to 20 per cent releases 96 per cent. But slowing only to 90 per cent releases just 19 per cent, and to 95 per cent barely 10 per cent.
This is the single most consequential design fact about flywheel storage, and it is where naive comparisons against batteries go wrong. A battery quoted at 10 kWh usually means something close to 10 kWh available. A flywheel quoted at 10 kWh of stored energy may deliver only 2 kWh if the driven equipment cannot tolerate a wide speed swing.
It also explains why modern flywheel systems sit behind power electronics rather than driving anything mechanically. A variable-frequency converter can accept a two-to-one or four-to-one speed change and still deliver constant voltage and frequency, which unlocks 75 to 94 per cent of the store. A directly coupled mechanical load rarely tolerates more than a few per cent, and strands most of the energy.
Standby Losses and Why They Set the Timescale
A flywheel loses energy continuously to bearing friction and aerodynamic drag, and drag is the dominant term at high rim speed because it grows roughly with the cube of speed. This is why serious flywheel systems run in a vacuum enclosure on magnetic bearings: at 500 metres per second, windage in air would empty the rotor in minutes.
Even a good system loses on the order of one to two per cent of its stored energy per hour. That makes a flywheel excellent for seconds-to-minutes duties — grid frequency regulation, uninterruptible power ride-through, smoothing a punch press or a lift — and unsuitable for storing energy overnight. Chemical and thermal stores lose far less over long periods; the flywheel wins on cycle life, response time and power density instead.
The comparison against field storage is worth making because the physics is so different. Our field energy density calculator shows that even a strong magnetic field stores only a few hundred kilojoules per cubic metre, whereas the defaults on this page store nearly seven megajoules in a rotor a metre across.
Where This Sits Next to the Other Rotation Tools
This page is the storage view of a rotor. For the general spinning-body case, including angular momentum and rolling motion, use the rotational kinetic energy calculator. To work out the moment of inertia of a shape that is not on the list here, use the moment of inertia calculator or the mass moment of inertia calculator.
For the speed conversions themselves, the angular velocity calculator handles rpm and radians per second, and the angular momentum calculator covers what is conserved when the rotor changes shape. The translational comparison is in the kinetic energy calculator.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Quoting stored energy as if it were deliverable — the usable figure depends entirely on the minimum working speed, and for a narrow speed range it can be a small fraction of the total.
- Using rpm in the energy formula — the expression needs radians per second. Forgetting the 2π/60 conversion changes the answer by a factor of about 96.
- Including shaft and housing in the rotor mass — only mass that spins at radius contributes to storage, and mass near the axis contributes almost nothing.
- Judging safety by rpm — rim speed and material density set the stress. A large wheel at modest rpm can be far closer to failure than a small one spinning much faster.
- Ignoring standby loss — a rotor in air at high rim speed loses its charge in minutes. Vacuum enclosures and magnetic bearings are what make the technology viable at all.
Related Free Tools From Arb Digital
The general spinning-body case is covered by the rotational kinetic energy calculator, and rotor geometry by the moment of inertia calculator and the mass moment of inertia calculator. Speed conversions live in the angular velocity calculator, conservation questions in the angular momentum calculator, and the linear analogue in the kinetic energy calculator. Rotor stress is handled by the hoop stress calculator and electromagnetic storage by the field energy density calculator. Everything Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
It is one half the moment of inertia times the square of the angular speed. For the defaults here, a 100 kg solid disc half a metre in radius at 10,000 rpm holds about 6.85 megajoules, which is 1.9 kilowatt-hours.
Because the rotor has to keep turning fast enough for the load or the power electronics to work. The recoverable fraction is one minus the square of the ratio of minimum to maximum speed, so a two-to-one speed range gives 75 per cent.
Yes, twice as much for the same mass and outer radius, because all of the mass sits at the largest radius. That is why storage flywheels use rims rather than solid wheels wherever the hub can be made light enough.
Rim speed, not rpm. Stress in the rotor goes as material density times rim speed squared, so the ceiling on specific energy is set by the strength-to-density ratio of the material, which is why composites are used.
It holds far less energy per kilogram but delivers it much faster, tolerates millions of cycles, and does not degrade with depth of discharge. Flywheels are chosen for power and cycle life, not for capacity.
Only minutes to hours in practice. Bearing friction and aerodynamic drag drain it continuously, so even vacuum-enclosed systems on magnetic bearings lose on the order of one to two per cent per hour.
It is the shape factor in the expression I equals k times mass times radius squared. A solid uniform disc has k of 0.5, a thin rim has 1, and a hub-mounted thick rim falls between the two.
Divide the usable energy by the time you need to deliver it over. That average power, not the energy itself, is what most flywheel systems are specified and purchased against.
This tool is provided for educational and study use. It models an ideal rigid rotor with no losses, no containment analysis and no material strength check, so treat its output as a physics teaching result rather than a basis for designing or operating rotating machinery.
