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What is the Conservation of Angular Momentum?

By H.R. Childress
Updated May 21, 2024
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The conservation of angular momentum is a fundamental concept in physics, along with the conservation of energy and the conservation of linear momentum. It states that the total angular momentum of a system must remain the same, which means it is conserved. Angular momentum is a vector property, which means that it is defined by both a magnitude and a direction, so the conservation of angular momentum also involves vectors.

Conservation of angular momentum applies to systems where the total torque applied is 0. Torque is rotational force, such as a twist. To determine if conservation of angular momentum applies, the sum of the angular momentums in the system before and after a change are summed together. If the angular momentum after the change minus that before the change equals 0, the angular momentum was conserved.

Angular momentum, often represented by the letter L in equations, is a property of the moment of inertia and the angular velocity of an object. The moment of inertia, usually represented by the letter I, is a measure of an object's resistance to changes in rotation. It is a function of the mass and shape of the object. The units of a moment of inertia are mass times area, but the exact formula for the moment of inertia depends on the shape of the object. Physics and engineering textbooks often include a chart with formulas for the moment of inertia of common object shapes to aid in calculations.

The angular velocity of an object is measured in radians per second and is usually represented by the Greek letter omega. It is calculated by dividing the component of the velocity vector that is perpendicular to the radius of motion by the radius. In practice, the result is often achieved by multiplying the magnitude of the velocity vector by the sine of the angle of the vector and dividing by the magnitude of the radius.

To find the angular momentum of an object, the moment of inertia is multiplied by the angular velocity. Since both are vector quantities, the conservation of angular momentum must also involve a vector quantity. Vector multiplication is performed to calculate the angular momentum, L = I*w.

If the object for which the angular momentum is being calculated is a very small particle, it can be calculated using the equation L = m*v*r. In this equation, m is the mass of the particle, v is the component of the velocity vector that is perpendicular to the radius of motion, and r is the length of the radius. The quantities in this equation are all scalar, and a positive sign or negative sign is used to indicate the direction of rotation.

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