What is the formula for angular momentum?

There are two formulas. For a rotating rigid body the angular momentum is the product of moment of inertia and angular velocity, so L = I*omega. For a point moving along a curved path, the angular momentum is the product of mass, velocity and radius, so L = m*v*r. Both formulas give the angular momentum in kg*m2/s and this calculator will pick the appropriate formula based on the values entered.

How do you calculate angular momentum given mass, velocity, and radius?

You multiply three quantities together: L = m*v*r, where m is the mass, v is the velocity along the path and r is the radius from the point of rotation to the object. For example, if a ball with a mass of 3 kilograms moves at a speed of 0.2 meters per second on a string that is 0.1 meters long, then its angular momentum will be equal to 3 multiplied by 0.2 and 0.1, which results in 0.06 kilogram-meter squared per second. When you enter the values for mass, velocity, and radius into their respective fields, the calculator will perform the multiplication.

What is Conservation of Angular Momentum?

This is a law which states that the angular momentum of a system remains constant if no external torques act on the system. When an ice skater pulls her arms in close to her body she speeds up her rotation because of this principle. As the moment of inertia decreases, the rotational speed must increase to keep the angular momentum constant. In the "Conservation" section of this calculator you can input the angular momentum and new moment of inertia to calculate the new rotational velocity.

What is the unit of angular momentum?

The unit of angular momentum in the International System of Units is kilogram-meter squared per second (kg·m2/s). This is the product of moment of inertia, expressed in kilogram-meters squared, and angular velocity, expressed in radians per second. The unit for angular momentum is the same as that for Planck's constant, which plays a central role in quantum mechanics.

What is the difference between angular momentum and moment of inertia?

The moment of inertia is a property of the object itself and describes how its mass is distributed around an axis of rotation. It does not change as long as the shape of the object remains the same. The angular momentum is the product of the moment of inertia and the angular velocity, so it also depends on the actual rotational speed of the object. A stationary flywheel has a moment of inertia but its initial angular momentum is zero.

Angular Momentum Calculator

Angular momentum

How do you want to work it out?

Pick what you know. Only the fields that method needs are shown.

Moment of inertia and angular velocity

Point particle: mass, speed and radius

Moment of inertia (I)

Angular velocity (ω)

Angular momentum (L)

kg·m²/s

Work the moment of inertia out from a shape

Turn a mass and a size into a moment of inertia for a sphere, disk, rod, ring or hoop.

Enter a moment of inertia and an angular velocity and the calculator finds the angular momentum from L = I × ω. Leave any one box blank to solve for it instead. Prefer to work with a point mass on a curved path? Open the mass, speed and radius section to use L = m × v × r. Not sure of the moment of inertia? The shape helper turns a mass and a size into I for you.

Please input values into the fields

Angular momentum is the equivalent of linear momentum but for rotational motion. While linear momentum represents how difficult it is to stop an object moving in a straight line, angular momentum describes how difficult it is to stop an object from rotating or orbiting.

This calculator offers two methods to determine the angular momentum and also allows for individual values to be calculated backwards. Leave one of the fields blank and it will automatically calculate that value.

What is angular momentum?

Angular momentum is denoted by the capital letter L and represents the amount of rotation an object has. It takes into account both the rotating mass and its distribution around the axis of rotation, as well as the overall speed of the rotation.

Angular momentum is a vector quantity, meaning it has both magnitude and direction. If you curl your right fingers in the direction of an object's rotation, then your thumb points in the direction of its angular momentum, which is perpendicular to the plane of rotation.

The SI unit is kilogram-meter squared per second (kg·m2/s). This may seem like a strange unit at first, but it comes directly from the formula. It is the product of moment of inertia with units kg·m2 and angular velocity with units radians per second.

Formula for angular momentum

Two basic formulas are covered here and both of them are integrated into this calculation tool. The first one is suitable for rigid bodies with extension such as wheels, discs or planets that rotate around their own axis.

L = I\,\omega

In this case I is the moment of inertia, i.e. the amount of mass involved in rotation. Omega is the angular velocity expressed as radians per second. The larger the moment of inertia and the faster the rotation, the greater the rotational momentum.

The second formula is suitable for point masses or small objects that move along a curved path while maintaining a constant distance from the axis of rotation. Imagine a ball hanging on a string or a planet moving in an orbit.

L = m\,v\,r

where m is the mass, v is the speed along the path and r is the radius, i.e. the distance of the object from the axis of rotation. These two formulas are consistent with each other. The moment of inertia for a point mass is the product of m and the square of r, while the angular velocity is equal to v divided by r. Therefore I*omega = m*v*r.

Moment of inertia and shape

The moment of inertia is what makes up most of the complexity. It depends not only on the mass of an object but also how far that mass is from the axis of rotation. For a given mass, having it spread out further away from the axis of rotation will make it harder to accelerate in rotation and give it more angular momentum.

Each standard shape has its own specific formula and this calculator applies those formulas for you. When you input mass and dimensions, it will give you the moment of inertia which can then be used in further calculations.

Shape	Axis	Moment of inertia
Solid disk or cylinder	Central axis	I = 1/2 · m · r²
Solid sphere	Through the centre	I = 2/5 · m · r²
Hollow sphere (thin shell)	Through the centre	I = 2/3 · m · r²
Thin ring, hoop or point mass	Central axis	I = m · r²
Thin rod	Through the centre	I = 1/12 · m · L²
Thin rod	Through one end	I = 1/3 · m · L²

Example calculation:

The moment of inertia of a wheel is 2 kilogram meters squared and the angular velocity is 1 radian per second. What is the angular momentum?

L = I\,\omega = 2 \times 1 = 2 \ \text{kg·m}^2/\text{s}

So the angular momentum of this wheel is 2 kilogram-meter squared per second. If you put 2 into the moment of inertia field and 1 into the angular velocity field, the calculator will calculate the angular momentum and give out 2.

In the case of a point mass, a ball with a mass of 3 kilograms is spinning on a string that's 0.1 meters long at a speed of 0.2 meters per second. Its angular momentum would be 0.06 kilogram-meter squared per second, which is equal to the product of 3, 0.2 and 0.1. You can check your answer by opening up the fields for mass, velocity, and radius, and entering those three values.

The law of conservation of angular momentum.

Angular momentum has a special property: as long as no external forces act on an object, its angular momentum remains constant. Physicists call this the law of conservation of angular momentum and consider it to be one of the most fundamental laws of mechanics.

A typical example is figure skaters. When they extend their arms, their moment of inertia is large and the rotation is slow. When they pull their arms in close to their body, the moment of inertia decreases so that they have to increase their rotational speed to keep the angular momentum constant.

I_1\,\omega_1 = I_2\,\omega_2 \qquad \omega_2 = \dfrac{L}{I_2}

If the moment of inertia is halved, then the rotational speed doubles. This calculator in the "Conservation" section will perform this calculation for you. If you enter the current angular momentum and new moment of inertia, it will give you the new rotational speed and new rotational energy.

The same law also explains why divers pull their bodies into a tuck position to spin faster, how cats manage to reorient themselves while falling, and why a collapsing star spins faster and becomes a fast-spinning neutron star.

Angular momentum and kinetic energy.

Rotating objects also store energy. The rotational kinetic energy is the product of the moment of inertia and the square of the angular velocity divided by two. It can also be expressed directly in terms of the angular momentum.

KE = \tfrac{1}{2} I \omega^2 = \tfrac{1}{2} L \omega

This calculator shows the angular momentum as well as the energy. Note that when a figure skater pulls her arms in close to her body to spin faster, the actual energy goes up even though the angular momentum stays constant. The extra energy comes from the work done by the muscles pulling the arms inward and resisting the rotation.

How to use this calculator:

Choose the method that matches your known information. For rigid body methods, enter the moment of inertia and angular velocity to read off the angular momentum. If any one of these three values is blank, it will be calculated for you.

For the point mass method open the Mass, Velocity and Radius fields and enter three of four values. If you do not know the moment of inertia required for the rigid body method then open the shape wizard, select a shape and enter the mass and dimensions.

Each field provides the opportunity to switch units. You can enter rotational speed in revolutions per minute (rpm) and read angular momentum in kilogram meters squared per second (kg m²/s), or you can input mass in pounds and radius in inches. The calculator will convert all values into consistent units before performing calculations.

Angular momentum units

The unit of angular momentum is the product of mass and length squared divided by time. In SI units it is kilogram meters squared per second (kg⋅m²/s). As angular momentum has the same dimension as Planck's constant, it is of great importance in quantum physics.

Quantity	Symbol	SI unit
Angular momentum	L	kilogram metre squared per second (kg·m²/s)
Moment of inertia	I	kilogram metre squared (kg·m²)
Angular velocity	ω	radian per second (rad/s); also rpm, rev/s
Mass	m	kilogram (kg)
Speed	v	metre per second (m/s)
Radius	r	metre (m)

Where does angular momentum occur?

Angular momentum can be found almost anywhere there is rotation. A spinning gyroscope can maintain its upright position because the angular momentum counteracts a force that would cause it to fall over. Gyroscopes used in smartphones and spacecraft use this same effect to detect and maintain orientation.

On the largest scale, angular momentum is what keeps planets in their orbits and galaxies spinning. On the smallest scale, electrons have a certain amount of angular momentum that determines the structure of every atom.

Common mistakes

The most common mistake is to confuse the units of angular velocity. There is a conversion factor between revolutions per minute and radians per second which is 2 pi divided by 60. So a wheel rotating at 3,000 rpm has an angular velocity of about 314 radians per second not 3,000. This calculator will do the conversion for you but you must specify the units that you are entering.

Another common problem is forgetting that the moment of inertia depends both on the object and the axis of rotation. For an identical rod, the moment of inertia about the end is four times as large as the moment of inertia about the center. Also remember that angular momentum is a vector. Negative values are not errors but indicate opposite directions of spin.

This tool is for general educational purposes and solving everyday problems. If used in research, engineering or safety-critical applications, please verify all numbers against the respective standards and data required by your project.

Frequently asked questions

What is the formula for angular momentum?: There are two formulas. For a rotating rigid body the angular momentum is the product of moment of inertia and angular velocity, so L = I*omega. For a point moving along a curved path, the angular momentum is the product of mass, velocity and radius, so L = m*v*r. Both formulas give the angular momentum in kg*m2/s and this calculator will pick the appropriate formula based on the values entered.
How do you calculate angular momentum given mass, velocity, and radius?: You multiply three quantities together: L = m*v*r, where m is the mass, v is the velocity along the path and r is the radius from the point of rotation to the object. For example, if a ball with a mass of 3 kilograms moves at a speed of 0.2 meters per second on a string that is 0.1 meters long, then its angular momentum will be equal to 3 multiplied by 0.2 and 0.1, which results in 0.06 kilogram-meter squared per second. When you enter the values for mass, velocity, and radius into their respective fields, the calculator will perform the multiplication.
What is Conservation of Angular Momentum?: This is a law which states that the angular momentum of a system remains constant if no external torques act on the system. When an ice skater pulls her arms in close to her body she speeds up her rotation because of this principle. As the moment of inertia decreases, the rotational speed must increase to keep the angular momentum constant. In the "Conservation" section of this calculator you can input the angular momentum and new moment of inertia to calculate the new rotational velocity.
What is the unit of angular momentum?: The unit of angular momentum in the International System of Units is kilogram-meter squared per second (kg·m2/s). This is the product of moment of inertia, expressed in kilogram-meters squared, and angular velocity, expressed in radians per second. The unit for angular momentum is the same as that for Planck's constant, which plays a central role in quantum mechanics.
What is the difference between angular momentum and moment of inertia?: The moment of inertia is a property of the object itself and describes how its mass is distributed around an axis of rotation. It does not change as long as the shape of the object remains the same. The angular momentum is the product of the moment of inertia and the angular velocity, so it also depends on the actual rotational speed of the object. A stationary flywheel has a moment of inertia but its initial angular momentum is zero.

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Angular Velocity CalculatorCalculate angular velocity (ω) from an angle and time, a linear speed and radius, a period or a frequency, and read it in rad/s, RPM, rev/s and deg/s. Solves every method backwards too.

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Angular Frequency CalculatorFind angular frequency (omega) from frequency, period, rpm, or an oscillating system, and get the matching frequency, period, wavelength, and peak motion.

Kinetic Energy CalculatorCalculate kinetic energy from mass and speed with KE = 1/2 m v squared, or solve for the mass or the speed. Includes momentum, the work-energy theorem, equivalent fall height, and the relativistic correction.

Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Angular momentum
Definition of angular momentum as a vector, the L = I omega and L = r x p forms, and the conservation law.
OpenStax University Physics, Angular Momentum
Angular momentum of a particle and of a rigid body, worked examples, and the connection to torque.
Wikipedia: Moment of inertia
Moment of inertia for standard shapes (sphere, cylinder, rod, ring) used in the shape helper, and its dependence on the axis.