Rotational Kinetic Energy Calculator

Rotational Kinetic Energy Calculator finds the energy of a spinning body from its moment of inertia and angular velocity, with the rolling total, stopping torque and average power.

Rotational Kinetic Energy Calculator: E = ½ I ω²

The Rotational Kinetic Energy Calculator finds the energy stored in a spinning body from its moment of inertia and its angular velocity. It can also work backward to the spin speed or the moment of inertia when the energy is known.

In simple terms, rotational kinetic energy is the energy a body has because it is turning. A wheel, a fan blade and a planet all carry it, even when their centers stay in one place. By definition it is the sum of the kinetic energies of every small piece of the body as it moves around the axis.

Adding up those pieces gives one compact formula. OpenStax University Physics derives it from the speed v = r ω of each piece.

$$E = \frac{1}{2} I \omega^2$$

Here I is the moment of inertia in kg·m² and ω is the angular velocity in rad/s. The symbol for the result varies by book. E, K, K_rot and KE_rot all mean the same quantity.

The unit used for rotational kinetic energy is the joule, the same as for any energy. Its dimensional formula is M L² T⁻², and it is a scalar, so it has a size but no direction. The calculator opens with I = 0.5 kg·m² at 10 rad/s, which gives 25 J.

Angular Velocity in rad/s, rpm or Hz

The formula needs angular velocity in radians per second. The Rotational Kinetic Energy Calculator also accepts rpm, revolutions per second and degrees per second, and converts them first. One rpm is π ÷ 30 rad/s, so 30 rpm is 3.142 rad/s.

A small wheel shows the steps. Take a 1 kg hoop of 0.5 m radius turning at 30 rpm. Its moment of inertia is 1 × 0.5², or 0.25 kg·m². The energy is ½ × 0.25 × 3.142², which comes to 1.234 J.

Speed matters far more than size, because it is squared. The “Energy at Other Speeds” panel of the Rotational Kinetic Energy Calculator shows this for the moment of inertia entered. These are its rows for the default 0.5 kg·m².

Doubling the speed stores four times the energy, and tripling it stores nine times. The fourth card states the same rule for the entry in use. For a speed given as a rim speed or a period, the Angular Velocity Calculator converts it to rad/s first.

Moment of Inertia From the Body List

The moment of inertia is often the unknown part. The Body list in the Rotational Kinetic Energy Calculator works it out for seven common bodies from the mass and its dimensions. The table uses a 4 kg body with a 0.5 m radius, or a 1 m length for the rods, spinning at 10 rad/s.

Mass placed far from the axis counts for more. The hoop stores twice the energy of the disk at the same mass, radius and speed. A rod spun about its end stores four times what it stores about its center. The thick ring sits between the disk and the hoop, and an inner radius of zero turns it back into the solid disk.

A textbook case checks the rod. OpenStax models four helicopter blades, each 4.00 m long and 50.0 kg, as rods turning about one end at 300 rpm. Enter 200 kg and 4 m with “Rod about one end” and the result is 526.4 kJ, the book’s 5.26 × 10⁵ J.

For any other shape, the Mass Moment of Inertia Calculator covers 23 shapes and offset axes. Leave Body on “Any body: enter I” and type that value in. Eight units are accepted, from kg·m² and g·mm² to lb·ft², oz·in² and slug·ft².

Rolling: Spin Plus Forward Motion

A rolling wheel has two kinds of kinetic energy at once. It spins, and its center also moves forward. OpenStax writes the total as the sum of both parts.

$$K = \frac{1}{2} m v^2 + \frac{1}{2} I \omega^2$$

When a wheel rolls without slipping, its forward speed is v = ω × r. For a round body from the list, the fourth card of the Rotational Kinetic Energy Calculator adds the two parts. The default 4 kg disk at 10 rad/s rolls at 5 m/s, with 25 J of spin and 50 J of forward motion, for 75 J in total.

The split depends only on the shape. Spin is one third of the total for a solid disk, half for a hoop, two sevenths for a solid sphere and two fifths for a hollow sphere. That is why a solid ball beats a hoop down a ramp.

These are two of the three formulas for kinetic energy that textbooks list. Motion in a line uses ½ m v², rotation uses ½ I ω², and speeds near that of light need the relativistic form. Calculating kinetic energy always starts with choosing which of these applies.

Solving for Angular Velocity or Inertia

The Calculate list in the Rotational Kinetic Energy Calculator turns the formula around. Enter a target energy in J, kJ, MJ, Wh, kWh, cal, kcal, ft·lbf or BTU, and the calculator solves for one of the other two quantities.

$$\omega = \sqrt{\frac{2 E}{I}}$$

With the default 60 J and I = 0.5 kg·m², the required angular velocity is 15.49 rad/s, or 147.9 rpm. Asked for the moment of inertia instead, 60 J at 10 rad/s needs 1.2 kg·m².

OpenStax gives a case that fits this mode. A string does 50 J of work on a pulley with I = 2.5 × 10⁻³ kg·m². Enter both and the Rotational Kinetic Energy Calculator returns 200 rad/s, the textbook answer.

Power, Torque and Stopping Time

Stored energy has to be put in or taken out, and that takes torque over time. The third card of the Rotational Kinetic Energy Calculator uses the Time for the Change from the options panel and assumes a steady change of speed.

For the default 25 J and a 2 s stop, the average power is 12.5 W and the torque is 2.5 N·m. The body makes 1.592 turns while it stops. The second card adds the angular momentum, 5 kg·m²/s, and the time per turn, 628.3 ms.

The formula for rotational power at any instant is torque times angular velocity. OpenStax states it for a constant net torque.

$$P = \tau \omega$$

At the start of the 2 s stop, that is 2.5 N·m × 10 rad/s, or 25 W. It falls to zero as the body slows, which is why the average is half as much. The other two common power formulas are work divided by time and force times velocity.

A body does not have to stop completely. The Final Speed field, which starts at 0, sets the speed it ends at. The energy change between two speeds follows from the same formula.

$$\Delta E = \frac{1}{2} I (\omega_2^2 – \omega_1^2)$$

Take the 0.68 kg·m² ring slowing from 10 rad/s to 4 rad/s in 2 s. The card leads with an Energy Lost of 28.56 J, at an average power of 14.28 W and a torque of 2.04 N·m. A final speed above the starting one turns the headline into Energy Gained.

The first card restates the energy in other units. The default 25 J is 18.44 ft·lbf, 6.944 mWh and 5.975 cal. It equals a 1 kg mass falling 2.549 m.

For the torque needed to reach a speed, the Angular Acceleration Calculator works from torque and inertia. The Angular Momentum Calculator handles L, and the Flywheel Energy Storage Calculator adds rim stress and usable energy for a storage wheel.

Rotational Potential Energy and Energy Levels

Rotation has a potential energy too. The rotational potential energy formula for a twisted shaft or torsion spring is ½ κ θ², where κ is the torsion constant in N·m per radian and θ is the twist angle. It trades back and forth with ½ I ω² in a torsional pendulum.

A body that swings under gravity stores potential energy in the height of its center of mass. The Physical Pendulum Calculator shows that swing energy for a rigid body.

A rotational energy level is the same idea at the scale of a molecule. A spinning molecule can only hold certain amounts of rotational energy, given by ħ² J (J + 1) ÷ (2 I) for whole numbers J. The moment of inertia I plays the same part there as in the formula above.

Kinetic Energy and Rotational Motion Questions

The questions below sit around the same topic. Each answer uses the definitions behind the Rotational Kinetic Energy Calculator.

What are the types of kinetic energy?

The two main types are translational, for motion along a path, and rotational, for spin. A third is vibrational, for back-and-forth motion inside a body or molecule. Lists of four usually add thermal energy, the random motion of molecules. A thrown baton is a good example, with 450 J of forward motion and 80.93 J of spin in the OpenStax case.

What are examples of rotational motion?

Five everyday examples are a ceiling fan, a car wheel, a spinning top, a washing machine drum and the Earth turning on its axis. A longer list adds a flywheel, a drill bit, a turbine, a propeller, a merry-go-round, a potter’s wheel, a record on a turntable, a hard disk, a gear, a pulley, a door on its hinges, a wrench on a bolt, a yo-yo, a figure skater and the hands of a clock.

What are the two types of rotational motion?

One is spin, where the axis passes through the body, as in a top. The other is revolution, where the body circles an axis outside itself, as the Earth does around the Sun. Rolling combines spin with forward motion.

What are the formulas for rotational kinematics?

For constant angular acceleration α, OpenStax lists ω = ω₀ + α t, θ = θ₀ + ω₀ t + ½ α t², and ω² = ω₀² + 2 α Δθ. The first of these is usually called the first equation of rotational motion. The Angular Displacement Calculator works with the angle.

Is rotational motion in NEET and JEE?

Both syllabi include it. JEE Main has a unit named Rotational Motion, with moment of inertia, radius of gyration, torque and angular momentum. NEET covers rotational motion under Motion of System of Particles and Rigid Body, including rolling. Skipping the chapter means giving up the questions set from it.

More tools for spinning bodies are on the Rotational Motion Calculators hub, next to the angular speed and circular motion tools.