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conservation of angular momentum

What is conservation of angular momentum?

Conservation of angular momentum is a physical property of a spinning system such that its spin remains constant unless it is acted upon by an external torque; put another way, the speed of rotation is constant as long as net torque is zero.

Angular momentum, sometimes referred to as spin, is determined by an object's mass, its velocity and how far the mass extends out from the point of rotation. The nearer the mass is to its axis point -- or the more consolidated it is around that axis -- the greater its velocity.

Angular momentum has its counterpart in linear momentum, which is defined as the product of mass and velocity. In other words, an object's momentum depends on both its mass and how fast it's moving. For example, a semitruck traveling at 25 mph has more momentum than a Mini Cooper traveling at the same speed. However, if the Mini collides into a fruit stand in front of a market, it will do far more damage at 25 mph than it would at 5 mph. The greater the mass or speed, the greater the momentum.

Angular momentum is similar to linear momentum, except that it also takes into account the distribution of mass around the point of rotation. In physics, it is defined as the product of rotational inertia and angular velocity:

  • Rotational inertia, also called moment of inertia, is concerned with an object's mass and its distribution. It is calculated by multiplying the mass by the radius squared. The radius is the distance from the object to its point of rotation.
  • Angular velocity refers to how quickly an object is rotating around its axis or traveling along a curved path.

The formula for angular momentum is written as L = Iω, where L is angular momentum, I is rotational inertia and ω (the Greek letter omega) is angular velocity. To simplify this, you can say that an object's angular momentum is the product of its mass, velocity and distance from the point of rotation. This formula is important to understanding the consequences of the conservation of angular momentum.

What is meant by conservation of angular momentum?

Like other aspects of physics, angular momentum is subject to the laws of conservation, which state that a specified property of a given physical system remains constant even as that system evolves over time. In other words, that property stays the same unless acted upon by an external force. The laws of conservation govern areas such as energy, electric charge, particle physics, linear momentum and angular momentum.

When applied to angular momentum, the law of conservation means that the momentum of a rotating object does not change unless some type of external torque is applied. Torque, in this sense, can refer to any outside force that acts upon the object to cause it to twist or rotate. Without the application of torque -- when net torque is zero -- the angular momentum remains constant; that is, the momentum remains conserved.

One example of this principle in action is a gyroscope, which exploits the law of conservation of angular momentum to stabilize, guide or measure rotational movement in many types of systems. The law of conservation also explains why a spinning Frisbee soars on a stable trajectory through the air rather than dropping immediately to the ground, or why a spinning top remains upright rather than submitting to gravity and toppling over.

Bicycle wheels also demonstrate the law of conservation in action. As the wheels spin, they behave like gyroscopes, generating their own angular momentum. The faster they spin, the greater the momentum and the greater the stability. If the wheels spin too slowly, the rider has a more difficult time maintaining equilibrium, in which case it will take only a small amount of torque to push that rider to the ground.

The law of conservation of angular momentum can be observed in other ways as well. For example, figure skaters, high divers and even people sitting in rotating chairs can demonstrate this law. If they spin with their arms stretched out, they'll continue to spin until an outside force is applied, such as the diver hitting the water. However, if they pull their arms in closer to their body, they'll spin even faster. This is because their angular momentum remains constant, but their mass distribution reduces in size, resulting in an increased velocity.

The law of conservation of angular momentum also applies to planets orbiting the sun. The closer a planet is to the sun, the greater its velocity. This holds true even if planets travel an elliptical orbit. As the distance of the object increases -- that is, as the planet moves farther away from the sun -- its velocity decreases, but as it moves closer, its velocity increases. However, the angular momentum remains constant.

Khan Academy provides a tutorial on the conservation of angular momentum in the following video.

See also: angular acceleration, gravitational acceleration, kinetic energy, locomotion, robotics, robot

This was last updated in June 2022

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