🔄 Rotational KE Calculator

KE = ½Iω² — Calculate rotational kinetic energy or solve for I or ω.

Solve for

Angular Velocity Input

Moment of Inertia I (kg·m²)

Angular Velocity (rad/s)

Rotational KE (J)

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Result

How to Use This Calculator

Choose what to solve for: rotational kinetic energy (J), moment of inertia (kg·m²), or angular velocity. Enter the other two values. You can input angular velocity in rad/s or in RPM using the toggle buttons, making it easy to work from motor datasheets that give speeds in RPM.

Select the solve mode from the dropdown. The most common is finding KE from I and ω.

Toggle between rad/s and RPM for the angular velocity input. The conversion is ω = RPM × 2π / 60.

Enter I in kg·m². If you do not know it, calculate it first using the Moment of Inertia calculator, then come back here.

Read the result in joules. The output also shows the equivalent in kilojoules for large energy values.

Rotational Kinetic Energy Formula

KE_rot = ½ × I × ω² ω (rad/s) = RPM × 2π / 60 Total KE (rolling object) = ½mv² + ½Iω² I = moment of inertia (kg·m²), ω = angular velocity (rad/s)

The formula is the rotational counterpart of KE = ½mv². Moment of inertia I plays the role of mass, and angular velocity ω plays the role of linear velocity. Doubling ω quadruples the stored energy, just as doubling speed quadruples linear kinetic energy.

Worked Examples

Flywheel: I = 0.5 kg·m², ω = 100 rad/sKE = ½ × 0.5 × 10,000 = 2,500 J

Motor disk: I = 0.02 kg·m², 3000 RPMω = 314.2 rad/s, KE = 988 J

Grinding wheel: I = 0.1 kg·m², 1500 RPMω = 157 rad/s, KE = 1,232 J

Basketball rolling at 5 m/s (solid sphere, m=0.62, r=0.12)KE_rot = 0.62 J, KE_total = 8.37 J

Where This Comes Up in Real Life

Flywheels store energy as rotational kinetic energy and release it smoothly. Some buses and trams use high-speed flywheel systems that spin at up to 60,000 RPM to recover braking energy. With a moment of inertia of 0.5 kg·m² at 6,283 rad/s (60,000 RPM), such a flywheel stores KE = ½ × 0.5 × 6,283² = 9.87 MJ, enough to propel a bus for about 500 metres.

Rolling objects have both translational and rotational kinetic energy. A solid sphere rolling without slipping uses KE_total = ½mv² + ½Iω² = ½mv² + ½(⅖MR²)(v/R)² = ⁷⁄₁₀mv². This means 28.6% of the kinetic energy is stored in the spin. A hollow sphere stores more in rotation (40%), which is why it reaches the bottom of a ramp later than a solid sphere of equal mass and radius.

Frequently Asked Questions

What is rotational kinetic energy?

KE_rot = ½Iω², where I is the moment of inertia (kg·m²) and ω is angular velocity (rad/s). It is the kinetic energy of a rotating body.

How does rotational KE differ from linear KE?

Linear KE = ½mv². Rotational KE = ½Iω². They are analogous: I replaces m, and ω replaces v. A rolling object has both.

What is a flywheel?

A flywheel is a rotating disk designed to store rotational kinetic energy. High I and high ω maximize stored energy. Used in engines and energy storage.

How do I convert RPM to rad/s?

ω (rad/s) = RPM × 2π / 60. For example, 1500 RPM = 1500 × 2π / 60 ≈ 157.1 rad/s.

What is total KE for a rolling object?

KE_total = ½mv² + ½Iω². For a solid cylinder rolling without slipping: KE_total = ¾mv².