Rotational Kinetic Energy Calculator
Calculate the energy stored in a spinning or rolling object from its moment of inertia, angular speed, mass and shape.
🌀 What is Rotational Kinetic Energy?
Rotational kinetic energy is the energy an object has because it spins. Every small piece of a rotating body is moving in a circle, so each piece carries kinetic energy, and adding them all up gives KE = ½ I ω². The quantity I is the moment of inertia, which measures how the mass is spread around the axis, and ω is the angular speed in radians per second. The formula has exactly the same shape as ½ m v², with the moment of inertia playing the role of mass and the angular speed playing the role of speed. This page computes that energy from I and ω, from a shape with a known mass and size, and for a body that rolls.
The idea matters wherever something heavy spins. A flywheel stores energy in its rim and gives it back when the load demands it. A spinning turbine, a car wheel, a washing machine drum, a hard disk platter and a figure skater in a spin all carry rotational energy. In sports a bowling ball, a bicycle wheel or a football rolling across the grass carries both energy of motion and energy of spin. Engineers use the formula to size flywheels and brakes, and students meet it in rolling problems, where a ball and a hoop released from the top of a ramp reach the bottom at different speeds because they split their energy differently.
Several details are easy to get wrong. The angular speed must be in radians per second, so a speed in rpm needs the factor 2π/60 before it goes into the formula. The moment of inertia depends on the axis as well as the shape, so the same rod has different energies when spun about its middle and about its end. A rolling object that does not slip is moving forward and spinning at once, and its total energy is the sum of the two parts, with the spin tied to the forward speed by ω = v / R. Finally, mass far from the axis counts for much more than mass near it, because the distance is squared.
This calculator also reports angular momentum L = Iω and draws a chart of energy against speed. The curve is a parabola, which shows how fast the stored energy rises when the object spins faster. For the straight-line counterpart, use the kinetic energy calculator, and for energy storage in a spinning rotor see the flywheel energy storage calculator.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — A Spinning Rotor
A rotor with I = 0.5 kg·m² spins at 120 rad/s
Example 2 — A Flywheel at 3000 rpm
A solid steel cylinder of mass 50 kg and radius 0.4 m spins at 3000 rpm
Example 3 — A Rod Spun About One End
A thin rod of mass 2 kg and length 1.5 m turns about one end at 10 rad/s
Example 4 — A Rolling Bowling Ball
A 6.35 kg ball of radius 0.109 m rolls without slipping at 8 m/s, treated as a solid sphere
Example 5 — A Rolling Hoop
A 2 kg hoop of radius 0.3 m rolls at 3 m/s
Example 6 — A Small Rotor at 6000 rpm
A rotor with I = 0.02 kg·m² spins at 6000 rpm
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for rotational kinetic energy?
Rotational kinetic energy is KE = 1/2 I w^2, where I is the moment of inertia in kg m^2 and w is the angular speed in rad/s. It is the rotational counterpart of 1/2 m v^2. A body with I = 0.5 kg m^2 spinning at 120 rad/s stores 3600 J.
How do you convert rpm to rad/s for rotational energy?
Multiply the rpm value by 2 pi and divide by 60. So 3000 rpm is 3000 x 2 pi / 60 = 314.16 rad/s. Only after this conversion can you use KE = 1/2 I w^2 and get joules.
What is the moment of inertia of a solid cylinder?
For a solid cylinder or disk spinning about its central axis, I = 1/2 m R^2. A 50 kg cylinder of radius 0.4 m has I = 0.5 x 50 x 0.16 = 4 kg m^2.
How much energy does a rolling ball have?
A solid sphere rolling without slipping has total kinetic energy 7/10 m v^2, which is 1/2 m v^2 for the motion plus 2/5 of that for the spin. A 6.35 kg ball at 8 m/s has 203.2 J of translational and 81.28 J of rotational energy, 284.48 J in total.
Why does a hoop have more rotational energy than a disk?
At the same mass, radius and angular speed, a hoop has I = m R^2 while a solid disk has I = 1/2 m R^2. All of the hoop's mass sits at the rim, so it stores twice the energy at the same spin rate.
Does rotational kinetic energy depend on the axis?
Yes. The moment of inertia changes with the axis, and so does the energy. A 2 kg, 1.5 m rod spinning at 10 rad/s has 18.75 J about its center but 75 J about one end.
What is the difference between rotational and translational kinetic energy?
Translational kinetic energy, 1/2 m v^2, belongs to the motion of the center of mass. Rotational kinetic energy, 1/2 I w^2, belongs to the spin about that center. A rolling wheel has both, and the total is their sum.
How is rotational kinetic energy related to angular momentum?
With angular momentum L = I w, the energy can be written KE = L^2 / (2 I). At fixed angular momentum, a smaller moment of inertia means a larger energy, which is why a skater who pulls in the arms speeds up and gains energy.
Can rotational kinetic energy be negative?
No. The angular speed is squared and the moment of inertia is positive, so the energy is never negative. The sign of the rotation direction does not matter for the energy, only for the angular momentum.
What units is rotational kinetic energy measured in?
Joules (J), the same as any other energy. With I in kg m^2 and w in rad/s, one kg m^2/s^2 equals one joule. The radian is dimensionless, so it drops out.
What fraction of a rolling object's energy is rotational?
The share is k / (1 + k), where k = I / (m R^2). That is 50 percent for a hoop, 33.3 percent for a solid cylinder, 28.6 percent for a solid sphere and 40 percent for a hollow sphere.