What is the primary difference between linear momentum and angular momentum?

The primary difference lies in the type of motion they describe. Linear momentum ($\vec{p} = m\vec{v}$) quantifies the inertia of an object in translational (straight-line) motion. It depends on mass and linear velocity. Angular momentum ($\vec{L}$) quantifies the inertia of an object in rotational motion. For a point particle, it's $\vec{r} \times \vec{p}$, and for a rigid body, it's $I\omega$. While both are vector quantities and represent 'momentum', angular momentum is always defined with respect to a specific origin or axis, making its direction dependent on the geometry of motion, unlike linear momentum which is independent of origin.

Why is angular momentum conserved when a figure skater pulls her arms in?

When a figure skater pulls her arms in, she reduces her body's moment of inertia ($I$). Moment of inertia is a measure of how mass is distributed relative to the axis of rotation; bringing mass closer to the axis decreases $I$. According to the principle of conservation of angular momentum, if no net external torque acts on the skater (which is largely true during her spin), her total angular momentum ($L = I\omega$) must remain constant. Since $I$ decreases, her angular velocity ($\omega$) must increase proportionally to keep the product $I\omega$ constant, causing her to spin faster.

Can angular momentum be conserved even if linear momentum is not?

Yes, absolutely. Consider a satellite orbiting the Earth in an elliptical path. The gravitational force from Earth acts on the satellite, providing a centripetal force. This force is always directed towards the center of the Earth. Since the force passes through the center of the Earth (which can be chosen as the origin), the torque due to this force about the Earth's center is zero ($\vec{\tau} = \vec{r} \times \vec{F}$, and $\vec{r}$ and $\vec{F}$ are anti-parallel or parallel). Therefore, the angular momentum of the satellite about the Earth's center is conserved. However, the satellite's linear momentum is continuously changing direction (and magnitude in an elliptical orbit), meaning its linear momentum is not conserved.

What is the significance of the cross product in the definition of angular momentum?

The cross product ($\vec{r} \times \vec{p}$) in the definition of angular momentum ($\vec{L}$) is significant because it inherently captures the 'rotational effectiveness' of the linear momentum. Only the component of linear momentum perpendicular to the position vector contributes to rotation. The cross product ensures that the resulting angular momentum vector is perpendicular to both $\vec{r}$ and $\vec{p}$, indicating the axis of rotation. Its magnitude ($rp\sin\theta$) highlights that maximum angular momentum occurs when $\vec{r}$ and $\vec{p}$ are perpendicular, and zero when they are parallel or anti-parallel (i.e., motion directly towards or away from the origin, which doesn't cause rotation about that origin).

How does angular momentum relate to Kepler's second law of planetary motion?

Kepler's second law states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This law is a direct consequence of the conservation of angular momentum. The gravitational force exerted by the Sun on a planet is always directed towards the Sun. If we choose the Sun as the origin, the torque due to this gravitational force about the Sun is zero. Consequently, the angular momentum of the planet about the Sun is conserved. The rate at which the area is swept out by the planet is directly proportional to its angular momentum. Since angular momentum is conserved, the rate of sweeping out area is also constant, which is precisely Kepler's second law.

Is angular momentum always conserved?

No, angular momentum is not always conserved. It is conserved only when the net external torque acting on the system is zero. If there is a net external torque, then the angular momentum of the system will change at a rate equal to that net external torque ($\vec{\tau}_{ext} = d\vec{L}/dt$). For example, if you apply a brake to a spinning wheel, the friction force creates an external torque that reduces the wheel's angular momentum. Similarly, air resistance can exert a torque on a spinning object, causing its angular momentum to decrease over time. The 'system' definition is also crucial; internal torques within a system do not change the total angular momentum of that system.

Angular Momentum

Physics

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Version 1Updated 22 Mar 2026

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Angular momentum, a fundamental vector quantity in physics, serves as the rotational analogue of linear momentum. For a point particle, it is defined as the cross product of its position vector relative to a chosen origin and its linear momentum vector. Mathematically, it is expressed as $\vec{L} = \vec{r} \times \vec{p}$ , where $\vec{r}$ is the position vector and $\vec{p}$ is the linear momentum…

Quick Summary

Angular momentum is the rotational equivalent of linear momentum, quantifying an object's 'spinning tendency'. For a point particle, it's defined as $\vec{L} = \vec{r} \times \vec{p}$ , where $\vec{r}$ is the position vector from a chosen origin and $\vec{p}$ is the linear momentum.

Its direction is given by the right-hand rule. For a rigid body rotating about a fixed axis, angular momentum simplifies to $L = I\omega$ , where $I$ is the moment of inertia and $\omega$ is the angular velocity.

The SI unit is J\cdot s or kg\cdot m^2/s. A crucial principle is the conservation of angular momentum: if the net external torque ( $\vec{\tau}_{ext}$ ) acting on a system is zero, its total angular momentum ( $\vec{L}_{total}$ ) remains constant.

This means $I\omega = \text{constant}$ for a rigid body. This principle explains phenomena like a figure skater speeding up when pulling in her arms or planetary motion. The rate of change of angular momentum is equal to the net external torque: $\vec{\tau}_{ext} = d\vec{L}/dt$ .

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Key Concepts

Angular Momentum of a Point Particle:

\vec{L} = \vec{r} \times \vec{p}

This formula is fundamental. It highlights that angular momentum is a vector quantity and its direction is…

Conservation of Angular Momentum:

I_1\omega_1 = I_2\omega_2

This principle is a direct consequence of Newton's second law for rotation ($\vec{\tau}_{ext} =…

Relation between Torque and Angular Momentum:

\vec{\tau} = d\vec{L}/dt

This equation is the rotational equivalent of Newton's second law ( $\vec{F} = d\vec{p}/dt$ ). It states that a…

Point Particle: —

\vec{L} = \vec{r} \times \vec{p} = \vec{r} \times (m\vec{v})

Magnitude (Point Particle): —

L = rp\sin\theta = rmv\sin\theta

Rigid Body (Fixed Axis): —

L = I\omega

Relation to Torque: —

\vec{\tau}_{ext} = \frac{d\vec{L}}{dt}

Conservation of Angular Momentum: — If

\vec{\tau}_{ext} = 0

, then

\vec{L}_{total} = \text{constant}

(i.e.,

I_1\omega_1 = I_2\omega_2

)

Units: — kg\cdot m^2/s or J\cdot s

Direction: — Right-hand rule for

\vec{r} \times \vec{p}

To remember the conservation of angular momentum: 'I Will Always Conserve'

I — Moment of Inertia

W — Angular Welocity (

\omega

)

A — Always

C — Conserve

This reminds you that $I\omega$ is conserved when external torque is zero. It's a simple way to recall the core principle $I_1\omega_1 = I_2\omega_2$ .