Angular momentum is a measure of the rotational motion of an object or system. It describes how much rotational motion an object has and how difficult that motion is to change. A spinning bicycle wheel, a rotating figure skater, a planet orbiting the Sun, and a satellite traveling around Earth all have angular momentum.
In physics, angular momentum plays a role similar to the role linear momentum plays in straight-line motion. Linear momentum describes motion along a path, while angular momentum describes motion around a point or axis. The concept is especially important because angular momentum is conserved in many physical systems, allowing scientists to explain and predict how rotating objects behave.
Understanding angular momentum requires three connected ideas: how it is calculated, how it changes when a force produces a turning effect, and why it remains constant when no external torque acts on a system.
What is angular momentum?
Angular momentum is a physical quantity associated with the motion of an object relative to a chosen reference point. For a rotating object, it depends on how much mass is involved, how that mass is distributed, and how quickly the object rotates. For an object moving along a path, angular momentum also depends on its position and direction of motion relative to the reference point.
The simplest way to picture angular momentum is to imagine a playground merry-go-round. When it spins, its mass moves around a central axis. The faster it rotates and the more mass it carries away from the axis, the greater its angular momentum, assuming the other factors remain unchanged.
Angular momentum is not limited to objects spinning around their own centers. A ball traveling in a circle around a central point has angular momentum relative to that point, even if the ball does not spin on its own axis. This distinction helps explain why planets, moons, and artificial satellites have angular momentum as they orbit other bodies.
Angular momentum is a vector quantity, meaning it has both magnitude and direction. For rotation around a fixed axis, its direction lies along that axis and follows the right-hand rule. If you curl the fingers of your right hand in the direction of rotation, your thumb points in the direction of the angular momentum vector.
The angular momentum formula
The appropriate angular momentum formula depends on whether an object is rotating around an axis or moving relative to a reference point.
For a rigid object rotating around a fixed axis, angular momentum is commonly expressed as:
L = Iω
Here, L is angular momentum, I is the moment of inertia, and ω is angular velocity.
Angular velocity describes how quickly an object changes its angular position. It is usually measured in radians per second. The moment of inertia describes how strongly an object’s mass distribution resists changes in its rotational motion. It depends not only on the object’s total mass but also on how far that mass lies from the axis of rotation.
For an object of fixed mass and shape rotating around a particular axis, increasing its angular velocity increases its angular momentum proportionally. Increasing its moment of inertia also increases its angular momentum at the same angular velocity.
For a particle moving relative to a reference point, the more general formula is:
L = r × p
In this equation, r is the position vector from the reference point to the particle, p is its linear momentum, and the symbol × represents the vector cross product.
Since linear momentum equals mass multiplied by velocity, the magnitude of angular momentum can also be written as:
L = mrv sin θ
Here, m is the particle’s mass, r is its distance from the reference point, v is its speed, and θ is the angle between the position and velocity vectors.
This equation shows that angular momentum depends on how the particle moves relative to the reference point. Motion directly toward or away from that point produces zero angular momentum because the position and velocity vectors point along the same line. Motion perpendicular to the position vector produces the greatest angular momentum for a given mass, distance, and speed.
The general vector formula applies to both straight-line and curved motion. The familiar expression L = Iω is a useful special case for a rigid body rotating around a fixed principal axis.
What is the moment of inertia?
The moment of inertia is one of the most important factors in understanding rotational motion. It measures how difficult it is to change an object’s angular velocity around a particular axis. It plays a role in rotational motion similar to the role mass plays in straight-line motion.
For a collection of particles, the moment of inertia is:
I = Σmᵢrᵢ²
In this equation, mᵢ represents the mass of each particle and rᵢ is that particle’s perpendicular distance from the rotation axis. The summation symbol means that the contributions of all the particles are added together.
The squared distance matters. A small amount of mass far from the axis can contribute substantially to the moment of inertia because its distance is squared. This is why a long rod can be harder to rotate about an axis through one end than about an axis through its center, even though its mass remains unchanged.
For example, consider a spinning office chair. If a person holds their arms close to their body, more of their mass lies near the rotation axis. If they extend their arms outward, their moment of inertia increases. At the same angular velocity, the extended-arm position corresponds to greater angular momentum.
Moment of inertia depends on the selected axis, not just the object’s shape and mass. The same object can have different moments of inertia when rotating around different axes. This makes the choice of rotation axis essential when calculating angular momentum.
How angular momentum changes: Torque
Angular momentum can change when an external torque acts on a system. Torque is the rotational effect of a force, describing how strongly that force tends to turn an object around a point or axis.
For a system of fixed mass and a fixed rotation axis, torque and angular momentum are related by:
τ = dL/dt
This equation states that the net external torque equals the rate of change of angular momentum. In situations where the angular momentum changes uniformly, the equation can be expressed as a change in angular momentum divided by the elapsed time.
Torque depends on the applied force and its lever arm, which is the perpendicular distance between the rotation axis and the force’s line of action. A force applied farther from a hinge generally produces more torque than the same force applied close to the hinge.
This principle explains why a door is easier to open when you push near its handle rather than close to its hinges. The force applied at the handle produces a larger torque because it acts at a greater distance from the axis of rotation.
A force does not necessarily produce torque. If its line of action passes directly through the rotation axis, its lever arm is zero, so it produces no torque about that axis. Likewise, a force directed entirely along the line connecting an object to a reference point produces no torque about that point.
Torque can increase or decrease angular momentum depending on its direction. A torque that acts in the direction of rotation can increase angular momentum, while a torque acting against the rotation can reduce it.
The law of conservation of angular momentum
The conservation of angular momentum states that the total angular momentum of a system remains constant when the net external torque acting on that system is zero.
Mathematically, this is expressed as:
Lᵢ = L𝒻
Here, Lᵢ is the initial angular momentum and L𝒻 is the final angular momentum. The equation applies to the total angular momentum of the system, provided no net external torque changes it.
Conservation of angular momentum is a fundamental principle of physics. It does not mean that every object must continue spinning at the same speed. Instead, the total angular momentum can remain constant even while the system’s parts change their speeds, positions, or rotational motions.
For a rigid body rotating around a fixed axis, the relationship L = Iω makes the consequences of conservation especially clear. If angular momentum remains constant, a change in moment of inertia must be accompanied by a corresponding change in angular velocity:
Iᵢωᵢ = I𝒻ω𝒻
If the moment of inertia decreases, the angular velocity increases. If the moment of inertia increases, the angular velocity decreases. This relationship assumes that the relevant angular momentum is conserved and that the rotation can be described using the same axis.
Conservation of angular momentum is closely related to the rotational symmetry of physical laws. In other words, the laws of physics do not change simply because an experiment is rotated to a different orientation in space. This deep connection is one reason angular momentum conservation is such a powerful principle throughout classical and modern physics.
In real situations, external torques may be present. Friction, air resistance, gravity, and contact forces can alter angular momentum depending on how they act on the system. Conservation remains applicable when the net external torque is zero or when external effects can be neglected over the period being considered.
Why a figure skater spins faster by pulling in their arms
A figure skater provides one of the clearest examples of angular momentum conservation.
When a skater spins with their arms extended, their mass is distributed relatively far from the rotation axis. This produces a comparatively large moment of inertia. When the skater pulls their arms inward, their moment of inertia decreases.
If the external torque on the skater is small, angular momentum remains approximately constant. Because angular momentum equals moment of inertia multiplied by angular velocity, the decrease in moment of inertia must be accompanied by an increase in angular velocity.
The skater therefore spins faster without needing an additional external force to accelerate the rotation. The change comes from redistributing mass relative to the rotation axis.
When the skater extends their arms again, their moment of inertia increases and their rotation slows. The total angular momentum can remain approximately constant throughout the process, even though the spinning speed changes considerably.
The skater’s rotational kinetic energy may also change. Pulling the arms inward requires muscular work, which can increase the rotational kinetic energy. Conservation of angular momentum does not mean that rotational energy must remain constant; these are different physical quantities governed by different conservation conditions.
Angular momentum in planetary orbits
Angular momentum is essential for understanding the motion of planets, moons, comets, and artificial satellites.
Consider a planet orbiting the Sun. The planet has angular momentum relative to the Sun because it has linear momentum and is located at a position that is generally not aligned with its direction of motion. Its angular momentum depends on its mass, orbital distance, speed, and direction of travel.
Gravity provides the force that changes the planet’s velocity and keeps it in orbit. However, in the idealized two-body model, the gravitational force between the Sun and the planet acts along the line connecting their centers. Because the force points directly toward or away from the reference point, it produces no torque about that point. The planet’s angular momentum relative to the Sun is therefore conserved.
This conservation law helps explain why a planet moves faster when it is closer to the Sun and slower when it is farther away. As the planet approaches the Sun, its orbital distance decreases. To preserve angular momentum, its tangential speed increases. As it moves farther away, its tangential speed decreases.
This behavior is summarized by Kepler’s second law: a line connecting a planet to the Sun sweeps out equal areas in equal intervals of time. The planet covers more distance along its orbit per unit time when it is closer to the Sun, compensating for its smaller orbital distance.
Real planetary systems contain multiple interacting bodies, so individual planets can exchange angular momentum through gravitational interactions. Nevertheless, the total angular momentum of an appropriately defined isolated system remains conserved.
Angular momentum in a bicycle wheel
A spinning bicycle wheel also illustrates angular momentum. As the wheel rotates, its mass moves around the axle, giving it angular momentum along the axle’s direction.
The wheel’s angular momentum increases with its rotational speed and depends on how its mass is distributed. A wheel with more of its mass concentrated near the rim has a larger moment of inertia than a wheel with the same mass concentrated closer to the axle.
A spinning wheel can resist changes in the direction of its rotation axis. This behavior is associated with gyroscopic effects, which arise from the relationship between angular momentum and torque. When an external torque acts on a spinning wheel, the response can involve a change in the direction of its angular momentum rather than simply a change in its rotational speed.
Gyroscopic behavior is used in navigation instruments, stabilization systems, and some motion sensors. It is also relevant to understanding bicycle dynamics, although a bicycle’s stability depends on several interacting factors, including steering geometry, the rider’s movements, and the motion of the wheels.
A spinning wheel does not maintain its orientation perfectly under all conditions. External torques can change its angular momentum, and friction gradually removes rotational energy. The key point is that changing the direction or magnitude of angular momentum requires a corresponding torque.
Angular momentum in collapsing stars
Angular momentum conservation also helps explain the behavior of stars as they evolve.
A star can rotate slowly while it is large. If the star’s core contracts substantially, its mass becomes concentrated closer to the rotation axis. This reduces the core’s moment of inertia. If little angular momentum is transferred out of the contracting region, its rotation rate increases.
The underlying principle is the same one that makes a figure skater spin faster when pulling in their arms. The scale is vastly different, but the relationship between mass distribution, moment of inertia, and angular velocity is the same.
This principle is especially relevant during the formation of compact astronomical objects, such as neutron stars. A contracting stellar core can spin much faster as it becomes smaller. However, the final rotation rate also depends on processes that transfer angular momentum, including magnetic interactions, interactions with surrounding material, and mass loss.
Angular momentum also influences the formation of planetary systems. A cloud of gas and dust can contract under gravity while rotating. As the material becomes more concentrated, conservation of angular momentum tends to increase its rotational speed. The resulting rotation helps shape the distribution of material in the developing system.
These examples demonstrate that angular momentum conservation operates across an extraordinary range of scales, from small mechanical devices to astronomical systems.
Angular momentum versus linear momentum
Angular momentum and linear momentum describe different aspects of motion, although they are closely related.
Linear momentum is defined as:
p = mv
Here, p is linear momentum, m is mass, and v is velocity. It describes an object’s translational motion, or movement from one location to another.
Angular momentum describes motion relative to a chosen reference point. For a particle, it is determined by the cross product of position and linear momentum:
L = r × p
This means an object moving in a straight line can have angular momentum if its path does not pass directly through the reference point. Conversely, an object moving directly toward or away from that point has zero angular momentum relative to it, even if it has substantial linear momentum.
The conservation laws also differ in their conditions. Linear momentum is conserved when the net external force on a system is zero. Angular momentum is conserved when the net external torque is zero.
The two quantities are related but cannot be substituted for one another. A moving object can have linear momentum without angular momentum relative to a particular point, and a rotating object can have angular momentum even when its center of mass remains stationary.
The units of angular momentum
In the International System of Units, angular momentum is measured in kilogram meters squared per second, written as kg·m²/s.
This unit follows from the formula L = Iω. The moment of inertia has units of kg·m², while angular velocity has units of s⁻¹. Multiplying them gives kg·m²/s.
The same unit follows from the particle formula L = r × p, because distance is measured in meters and linear momentum in kg·m/s.
Angular momentum is distinct from energy, which is measured in joules, and torque, which is measured in newton-meters. Torque and energy share the same combination of base units, but they represent different physical quantities. Torque describes the turning effect of a force, while energy measures the capacity to do work or produce physical change.
Why angular momentum matters
Angular momentum provides a unified way to understand rotational motion across physics. It explains why a skater spins faster when pulling in their arms, why a planet speeds up as it approaches the Sun, and why contracting astronomical objects can rotate much faster than their original forms.
Its mathematical relationships connect mass distribution, rotational speed, force, and motion. The formula L = Iω describes the angular momentum of a rigid body rotating around a fixed axis, while L = r × p expresses the more general relationship between position and linear momentum.
The conservation principle is especially valuable because it allows scientists to predict how a system will behave even when the detailed forces and motions of its components are complicated. When the net external torque is zero, the total angular momentum remains constant, although the distribution of that angular momentum among the system’s parts can change.
From everyday rotating objects to planetary orbits and stellar evolution, angular momentum is one of the fundamental tools for understanding how the physical world moves.
