Flywheel Energy Storage - Stored kJ, kWh, and discharge time

A flywheel energy storage calculator turns a rotor's mass, radius, geometry, and spin speed into the kinetic energy stored in the spinning mass through E = 1/2 I omega^2. The same inputs give rim speed and discharge time at a chosen load, useful for flywheel UPS sizing, KERS studies, and rotational dynamics homework.

• • Flywheel UPS Sizing: Pick a rotor mass, radius, and rpm that store enough kWh to ride through a grid outage.
• • KERS and Regen Braking: Estimate how much kinetic energy a spinning mass can absorb from a braking event.
• • Rotational Dynamics Homework: Plug m, r, and rpm into E = 1/2 I omega^2 and read J, kWh, omega, and rim speed.
• • Composite Rotor Trade Studies: Compare a thin-rim hoop against a hollow ring of the same mass and radius.

The flywheel stores energy as rotational kinetic energy, so the answer scales with the square of the spin speed. Doubling rpm multiplies stored energy by four.

For a flywheel whose spin angular momentum L = I omega also matters, the Angular Momentum Calculator returns L in kg m^2/s alongside the same kinetic energy.

The flywheel energy storage calculator picks the moment of inertia from the rotor geometry, converts rpm to rad/s, and applies E = 1/2 I omega^2.

• E: Stored rotational kinetic energy in joules.
• I: Moment of inertia about the spin axis in kg m^2. Shape dependent.
• omega: Angular velocity in radians per second (rpm times 2 pi / 60).
• m: Rotor mass in kilograms.
• r: Outer radius of the rotor in metres.
• r_i: Inner radius of a hollow ring rotor in metres (only used for ring geometry).
• v: Rim speed in m/s, equal to r times omega.

The calculator reads the shape selector to pick the moment-of-inertia formula: solid disk and cylinder use I = 0.5 m r^2, thin rim uses I = m r^2, hollow ring uses I = 0.5 m (r_o^2 + r_i^2). The rpm reading is converted to rad/s as omega = rpm * 2 pi / 60.

Rim speed v = r * omega is the practical ceiling on a flywheel. Steel rims are rated 400 to 600 m/s, carbon-fibre composite rims reach 600 to 1000 m/s. Past those limits, the rotor must spin slower, use a stronger material, or use a larger radius.

According to HyperPhysics, K = 1/2 I omega^2 and the moment of inertia of a solid disk is I = 1/2 m r^2.

According to NIST SP 811, angular velocity is measured in radians per second with 1 revolution equal to 2 pi radians, so omega = rpm * 2 pi / 60.

Because the moment of inertia I drives every answer here, the Moment of Inertia Calculator returns closed-form I values for common rotor shapes.

Four ideas sit underneath every flywheel answer: the moment of inertia that captures geometry, the angular velocity that captures the spin, the rotational kinetic energy formula E = 1/2 I omega^2, and the rim speed that sets the safety limit.

These four ideas are linked: changing geometry changes I for the same mass, changing rpm changes omega and so changes E quadratically, and the same omega produces different rim speeds at different radii.

Keeping them separate in the UI lets the same tool serve a student working through E = 0.5 I omega^2 and a designer checking the rim-speed limit.

Since angular velocity omega appears in E = 1/2 I omega^2 as a square, the Angular Velocity Calculator converts between rpm, hertz, and rad/s for the same motion.

Pick the rotor geometry, enter mass, outer radius, inner radius if applicable, spin speed in rpm, and discharge load in kW, then read the results in one screen.

1. 1 Select the rotor geometry: Choose Solid Disk, Thin Rim, Solid Cylinder, or Hollow Ring so the moment of inertia matches the rotor.
2. 2 Enter mass and outer radius: Type the rotor mass in kilograms and the outer radius in metres. Hollow ring also takes an inner radius.
3. 3 Enter the spin speed in rpm: The calculator converts it to rad/s as omega = rpm * 2 pi / 60.
4. 4 Enter the discharge load in kW: Type the constant electrical load. Leave at zero if you only want stored energy and rim speed.
5. 5 Read stored energy and rim speed: Read E in J and kWh on the result card, then check the rim speed in m/s against the material limit.
6. 6 Read the discharge time: Seconds the flywheel can sustain the chosen load under the idealised constant-power assumption.

For a 50 kg steel disk at 0.3 m radius spinning at 6000 rpm with a 5 kW load, the calculator returns I = 2.25 kg m^2, omega = 628.32 rad/s, E = 444,132 J (0.1234 kWh), v = 188.50 m/s, discharge time 88.83 s.

For the same motion expressed in work and power, the Work Energy Power Calculator converts the stored energy into work and power outputs.

This calculator keeps every important output on one screen so the same motion stays consistent across kinetic energy, rim speed, and runtime calculations.

• • Four rotor geometries in one place: Solid disk, thin rim, solid cylinder, and hollow ring rotors, so I matches the geometry.
• • Both J and kWh on the same card: Stored energy in joules and kilowatt-hours for homework and UPS sizing.
• • Automatic rpm to rad/s conversion: Inputs stay in rpm; result is in SI joules.
• • Rim speed check built in: Spot a rotor that would tear itself apart before committing to a design.
• • Discharge time at a chosen load: A runtime in seconds, which a UPS or KERS study needs.

These benefits matter most when the same rotor design answers both a physics problem and a sizing problem. The SI number drops into a battery comparison or KERS calculation without unit rework.

When the flywheel transfers its angular momentum to a coupled load, the Conservation of Momentum Calculator checks conservation of angular momentum across the transfer.

Four inputs drive every flywheel answer. The cards below explain how each input changes the result.

• • The discharge time assumes a constant electrical load and ignores bearing friction, windage, and eddy-current losses, so real runtime is shorter than reported.
• • The moment of inertia formulas assume uniform density and a rotor spinning about its geometric symmetry axis. Composite rotors with rim mass are approximated by the thin-rim formula.
• • The rim-speed limit is a material guideline, not a structural calculation. Real designs check hoop stress and burst speed separately.

These factors explain why two flywheels with the same mass can store different amounts of energy: mass at a larger radius or higher rpm wins, as long as the rim-speed limit is respected.

According to Wikipedia — Flywheel Energy Storage, modern composite flywheels reach rim speeds above 600 m/s and store tens of kilowatt-hours in rotors with rim-tip velocities of several hundred metres per second.

Once the calculator returns omega and the discharge load in kW, the Torque, Power and Speed Calculator converts the same omega and torque into mechanical power in watts.