Rotational Kinetic Energy Calculator
Enter the moment of inertia and angular velocity of any spinning object to find its rotational kinetic energy. Choose from eight common rigid-body shapes to let the calculator derive the moment of inertia from mass and size, or enter a custom value directly. Angular velocity accepts rad/s, RPM, or Hz, and results are shown in joules, kilojoules, or foot-pounds. The step-by-step panel shows every arithmetic operation with your real numbers.
What is rotational kinetic energy?
Rotational kinetic energy is the energy a rigid body possesses because it is spinning. It is the rotational analogue of the familiar translational kinetic energy (0.5 * m * v^2). Instead of mass, the relevant quantity is the moment of inertia I, which measures how the mass is distributed relative to the rotation axis. Instead of linear velocity v, the relevant quantity is the angular velocity omega in radians per second. The formula is KE = 0.5 * I * omega^2. When an object rolls without slipping, such as a ball rolling down a ramp, the total kinetic energy is the sum of its translational and rotational parts.
Moment of inertia: why it matters
The moment of inertia (I, measured in kg*m^2) plays the same role for rotation that mass plays for linear motion. A high moment of inertia means the object resists changes in spin, which is why a spinning flywheel stores a lot of energy and is hard to stop. The value depends not just on how much mass the object has but on where that mass sits relative to the rotation axis. Concentrating mass near the rim (as in a thin ring or hollow cylinder) gives a much higher I than concentrating it at the center (as in a solid disk of the same mass and radius). For a solid disk, I = 0.5 * m * r^2; for a thin ring all mass at radius r, I = m * r^2, exactly twice as much. This is why a hollow cylinder reaches the bottom of a ramp more slowly than a solid one: more of its energy is in rotation.
Unit conversions and practical usage
In SI units, energy is measured in joules (J), moment of inertia in kg*m^2, and angular velocity in radians per second. In practice, rotation speeds are often quoted in RPM (revolutions per minute): to convert, multiply by 2*pi/60, giving approximately 0.10472 rad/s per RPM. A motor spinning at 1800 RPM has omega = 1800 * 2*pi/60 = 188.5 rad/s. Energy can also be expressed in kilojoules, kilowatt-hours, or foot-pounds depending on engineering context. Flywheel energy storage, for example, is usually quoted in kWh, while ballistic and structural calculations often use ft*lbf.
Angular momentum and the spinning-up equation
Closely related to kinetic energy is angular momentum, L = I * omega (units: kg*m^2/s). Angular momentum is conserved when no external torque acts, which explains why a figure skater spins faster after pulling their arms in: I decreases, so omega must increase to keep L constant, and the kinetic energy actually rises (the extra energy comes from the muscular work of pulling the arms in). To spin a stationary object up to angular velocity omega, the required torque multiplied by spin-up time equals the angular impulse, which equals the final angular momentum. Knowing the kinetic energy also tells you the average power needed: power = KE / spin-up time.
Moment of inertia formulas for common shapes
| Shape | Axis | Formula | Example (m=1 kg, r=0.1 m) |
|---|---|---|---|
| Solid sphere | Center | I = 2/5 * m * r^2 | 0.004 kg*m^2 |
| Hollow sphere (shell) | Center | I = 2/3 * m * r^2 | 0.00667 kg*m^2 |
| Solid cylinder/disk | Central axis | I = 1/2 * m * r^2 | 0.005 kg*m^2 |
| Hollow cylinder | Central axis | I = 1/2 * m * (r1^2 + r2^2) | depends on r1, r2 |
| Thin ring | Central axis | I = m * r^2 | 0.01 kg*m^2 |
| Thin rod (center) | Perpendicular | I = 1/12 * m * L^2 | 0.000833 kg*m^2 (L=0.1) |
| Thin rod (end) | Perpendicular | I = 1/3 * m * L^2 | 0.00333 kg*m^2 (L=0.1) |
These are the standard formulas for the moment of inertia of uniform, rigid bodies rotating about their symmetry axis. m = mass (kg), r = radius (m), L = length (m).
Frequently asked questions
What is the formula for rotational kinetic energy?
The formula is KE = 0.5 * I * omega^2, where I is the moment of inertia in kg*m^2 and omega is the angular velocity in radians per second. This mirrors the translational formula KE = 0.5 * m * v^2 with mass replaced by moment of inertia and linear speed replaced by angular speed.
How do I convert RPM to rad/s?
Multiply by 2*pi/60, which is approximately 0.10472. So 300 RPM = 300 * 0.10472 = 31.42 rad/s. The calculator does this conversion automatically when you select RPM as the input unit.
How is the moment of inertia different for a solid cylinder vs a hollow cylinder?
A solid cylinder of mass m and radius r has I = 0.5 * m * r^2. A hollow cylinder (thin wall) of the same mass and outer radius has I = m * r^2, which is twice as large. The hollow version resists spin changes more because all the mass is far from the axis. A thick-walled hollow cylinder with inner radius r1 and outer radius r2 falls in between: I = 0.5 * m * (r1^2 + r2^2).
Why does a hollow cylinder roll down a slope more slowly than a solid one?
For a given total mass and radius, the hollow cylinder has a higher moment of inertia. When rolling without slipping, the kinetic energy is split between translational (0.5 * m * v^2) and rotational (0.5 * I * omega^2) components. The hollow cylinder stores a larger fraction of the available gravitational potential energy in rotation, leaving less for translation, so it accelerates more slowly.
What is angular momentum and how is it related to rotational KE?
Angular momentum L = I * omega. It is related to KE by KE = L^2 / (2 * I) or equivalently KE = 0.5 * L * omega. Unlike energy, angular momentum is conserved when no external torque acts on the system. Reducing the moment of inertia (for example, a skater pulling in their arms) increases omega proportionally, keeping L the same but increasing the kinetic energy.