# Rotational Kinetic Energy Calculator (KE = ½Iω²)

Calculate rotational kinetic energy with KE = 1/2 Iω². Enter moment of inertia and angular velocity in rad/s or RPM to find energy in Joules.

## What this calculates

Rotational kinetic energy is the energy an object has because it is spinning. A figure skater pulling their arms in spins faster and stores more rotational energy without any external push. This calculator finds that energy from the moment of inertia and angular velocity, with built-in RPM to rad/s conversion.

## Inputs

- **Moment of Inertia (I)** (kg*m^2) — min 0
- **Angular Velocity Input** — options: rad/s, RPM
- **Angular Velocity** (rad/s or RPM) — min 0

## Outputs

- **Angular Velocity** (rad/s) — Angular velocity in radians per second
- **Angular Velocity** (RPM) — Angular velocity in revolutions per minute
- **Rotational Kinetic Energy** (J) — Kinetic energy of rotation in Joules
- **Rotational Kinetic Energy** (kJ) — Kinetic energy in kilojoules

## Details

## How Rotational Kinetic Energy Works

The formula: **KE = 1/2 I omega squared**, where I is the moment of inertia (how mass is distributed around the axis) and omega is the angular velocity in radians per second.

### Converting RPM to rad/s

Most real-world specs list rotation speed in RPM. To convert: **omega = RPM x 2 pi / 60**. A car engine at 3,000 RPM spins at about 314 rad/s.

### Moment of Inertia Quick Reference

| Shape | Formula |
|-------|---------|
| Solid cylinder/disk | I = 1/2 mr squared |
| Hollow cylinder | I = m r squared |
| Solid sphere | I = 2/5 mr squared |
| Thin rod (center) | I = 1/12 mL squared |

### Real-World Examples

- **Flywheel energy storage:** Industrial flywheels spin at 10,000+ RPM to store energy for grid stabilization.
- **Bicycle wheel:** A 1.5 kg wheel with r = 0.35 m at 200 RPM stores about 13 J of rotational energy.
- **Washing machine drum:** The spin cycle at 1,200 RPM uses rotational kinetic energy to extract water from clothes.

## Frequently Asked Questions

**Q: What is moment of inertia?**

A: Moment of inertia measures how hard it is to change an object's rotation. It depends on both the mass and how that mass is spread out from the spin axis. A hollow ring has more moment of inertia than a solid disk of the same mass and radius because all the mass sits far from the center.

**Q: Why does angular velocity use radians per second?**

A: Radians are the natural unit for rotation in physics because they simplify formulas. One full revolution equals 2 pi radians (about 6.28). Using radians means KE = 1/2 I omega squared works directly without extra conversion factors. RPM is more intuitive for everyday use, which is why this calculator converts between the two.

**Q: How does rotational kinetic energy differ from linear kinetic energy?**

A: Linear kinetic energy uses mass and velocity (KE = 1/2 mv squared), while rotational kinetic energy uses moment of inertia and angular velocity (KE = 1/2 I omega squared). A rolling object like a bowling ball has both types simultaneously. The total kinetic energy is the sum of both.

**Q: Can I use this for flywheels and energy storage?**

A: Yes. Flywheel energy storage systems are a direct application of this formula. A flywheel with I = 5 kg m squared spinning at 10,000 RPM stores about 2.74 MJ of energy. Modern flywheels use carbon fiber composites to maximize the speed-to-mass ratio for grid-scale energy storage.

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Source: https://vastcalc.com/calculators/physics/rotational-kinetic-energy
Category: Physics
Last updated: 2026-04-08