Angular Momentum Calculator - L for Rigid Bodies and Particles

An angular momentum calculator returns the scalar magnitude of angular momentum L from either a rigid body's moment of inertia and angular velocity or a point particle's mass, speed, radius, and angle, then reports the matching kinetic energy and rotation period. The primary answer is given in kg m squared per second so it drops directly into torque, gyroscope, and orbital mechanics problems.

• • Rigid Body Spin: Compute L = I omega for a flywheel, turbine, or disk using its moment of inertia and spin rate.
• • Orbital Motion: Compute L = m v r sin theta for a satellite, planet, or electron orbiting a much heavier central mass.
• • Conservation Checks: Compare initial and final L across a torque-free event such as an ice skater pulling in their arms or a satellite changing orbit.
• • Right-Hand Rule Sign: Pair the scalar magnitude with the right-hand rule to assign a vector direction along the rotation axis.

The two governing formulas are L = I omega for a rigid body about a symmetry axis and L = m v r sin theta for a single particle. Angular momentum is conserved whenever the net external torque is zero.

When the rigid-body side of L = I omega is the focus, the Moment Of Inertia Calculator gives the closed-form I for common shapes so the same I value can be reused here.

The angular momentum calculator reads the mode selector and the formula-specific inputs, applies L = I omega for rigid bodies or L = m v r sin theta for particles, and derives the kinetic energy, angular velocity, rotation period, and rotation rate from the same motion.

• L: Angular momentum magnitude in kilogram metres squared per second.
• I: Moment of inertia of the rigid body about the rotation axis in kg m^2.
• omega: Angular velocity of the rigid body, or the effective angular velocity v sin theta / r for a particle.
• m: Particle mass in kilograms.
• v: Linear speed of the particle in metres per second.
• r: Perpendicular distance from the rotation point to the line of motion in metres.
• theta: Angle between the position vector r and the velocity vector v, in degrees or radians.

In rigid body mode, the calculator multiplies I by omega and computes kinetic energy as 0.5 * I * omega^2, the period as 2 pi / omega, and the rate as omega * 60 / (2 pi).

In point particle mode, the calculator converts the angle to radians, applies L = m * v * r * sin theta, and uses the same sin theta to derive the effective angular velocity omega = v sin theta / r. The kinetic energy is 0.5 * m * v^2 * sin^2 theta, which is the rotational share of the linear kinetic energy.

According to OpenStax University Physics Volume 1, the angular momentum of a point particle is the cross product L = r x p with magnitude m v r sin theta, and for a symmetric rigid body it reduces to L = I omega where I is the moment of inertia about the rotation axis

According to NASA JPL Planetary Fact Sheet, Earth orbits the Sun with mass 5.9722 x 10^24 kg, mean orbital speed 29780 m/s, and mean orbital radius 1.496 x 10^11 m

Because the particle form L = r x p is the magnitude of a cross product, the Cross Product Calculator is the right place to step through the vector that produces m v r sin theta.

Four ideas make every angular momentum calculator result easier to read: the position vector r, the linear momentum p, the moment of inertia I, and the right-hand rule that turns the scalar magnitude into a vector.

The cross product L = r x p is the general form that the magnitude formula m v r sin theta comes from. Whenever r and v meet at an angle other than 90 degrees, the sin theta factor is what makes the answer come out right.

Since omega in L = I omega is the same angular frequency that drives SHM and RLC arguments, the Angular Frequency Calculator converts between hertz, period, and rad/s for the same omega.

Pick the mode that matches the inputs you already have, fill in the formula-specific row, and read L alongside the kinetic energy, rotation period, and rotation rate.

1. 1 Select the calculation mode: Use Rigid Body when you know the moment of inertia and angular velocity, or Point Particle when you know the mass, speed, radius, and angle.
2. 2 Enter the rigid body inputs: Type the moment of inertia in kg m^2 and the angular velocity in rad/s. Solid disks, hoops, rods, and spheres have closed-form I values.
3. 3 Or enter the point particle inputs: Type the mass in kg, the speed in m/s, the radius in m, and the angle theta in degrees. For a circular orbit, theta is 90 degrees and the formula simplifies to L = m v r.
4. 4 Read the angular momentum: The primary card shows L in kg m^2/s. The other rows give the rotational kinetic energy, angular velocity, rotation period, and rotation rate.
5. 5 Switch modes for a cross-check: If both descriptions apply, switch the mode selector, re-enter the matching values, and verify that L comes out the same.

For Earth orbiting the Sun, switch to Point Particle, enter m = 5.9722 x 10^24 kg, v = 29780 m/s, r = 1.496 x 10^11 m, and theta = 90 deg, then read L = 2.661 x 10^40 kg m^2/s.

When the geometry behind r and theta is the next step, the Rotation Calculator rotates a 2D point by the same angle used in the L = m v r sin theta formula.

The angular momentum calculator keeps both angular momentum formulas in one place, applies the right units automatically, and exposes the kinetic energy and rotation period so the same motion stays consistent across every step.

• • Two formulas in one place: Switch between L = I omega for rigid bodies and L = m v r sin theta for particles without leaving the page.
• • SI units handled automatically: Inputs and outputs stay in kg, m, s, and rad so the result reads as kg m^2/s without manual conversion.
• • Built-in energy and period: Rotational kinetic energy, angular velocity, period, and RPM come from the same motion so conservation checks are immediate.
• • Right-hand rule ready: The scalar magnitude sits next to a constant reminder of the right-hand rule so the vector direction can be added in by hand for precession and gyroscope problems.

When the answer disagrees with an expectation, check whether the mode selector matches the inputs, whether theta is 90 degrees or another value, and whether the radius is the perpendicular distance from the axis or the total distance to the particle.

Once L and omega are known, the rotational power P = tau * omega is the next quantity, and the Torque to Horsepower Converter handles that torque-to-horsepower step without leaving the rotation topic.

Angular momentum scales with every input in a specific way, so the four cards below show how each input shifts L, the kinetic energy, and the rotation rate.

• • The calculator reports the scalar magnitude of angular momentum. Vector addition with the right-hand rule has to be done by hand for problems with two or more simultaneous rotations.
• • Inputs are limited to SI units of kg, m, s, and rad/s. Imperial pounds, feet, and RPM have to be converted before the result is read in kg m^2/s.

For a point particle, the radius r is the perpendicular distance from the rotation axis to the particle's line of motion, not the total distance from the origin. Using the total distance overestimates L by a factor of 1 / sin theta whenever theta is not 90 degrees.

According to NIST Guide for the Use of the SI, the SI unit of angular momentum is the kilogram metre squared per second, written kg m^2/s, and is obtained from mass times length squared per time

When the same omega is later used in a wave equation, the Harmonic Wave Equation Calculator plugs the rad/s value into a harmonic waveform problem with consistent units.