Conservation of Angular Momentum

For a point mass $m$ moving with velocity $\vec{v}$ at position $\vec{r}$ from a reference point, angular momentum is defined as $\vec{L} = \vec{r} \times \vec{p} = \vec{r} \times m\vec{v}$. The magnitude is $L = rmv\sin\theta$, where $\theta$ is the angle between $\vec{r}$ and $\vec{v}$. For a rigid body rotating about a fixed axis, this simplifies to $L = I\omega$, where $I$ is the moment of inertia and $\omega$ is the angular velocity. The SI unit of angular momentum is kg·m²/s.

Newton's second law for rotation states that the net external torque equals the rate of change of angular momentum: $\vec{\tau}_{net} = \frac{d\vec{L}}{dt}$. This is the rotational analogue of $\vec{F}_{net} = \frac{d\vec{p}}{dt}$. Notice that internal torques — forces that system parts exert on each other — always come in equal and opposite pairs (Newton's third law), so they cancel out when summing over the whole system. Only external torques can change the total angular momentum of a system.

If the net external torque on a system is zero ($\vec{\tau}_{net} = 0$), then $\frac{d\vec{L}}{dt} = 0$, which means $\vec{L} =$ constant. Written explicitly: $\vec{L}_i = \vec{L}_f$, or $I_i\omega_i = I_f\omega_f$. This is the conservation of angular momentum. Note that $\vec{\tau}_{net} = 0$ can occur either because there are no external torques at all, or because all external torques cancel each other. The conservation applies to each component of $\vec{L}$ independently along any axis where torque is absent.

Unlike mass in linear momentum, the moment of inertia $I = \sum m_i r_i^2$ of a system can change if the distribution of mass relative to the rotation axis changes. When a skater pulls their arms inward, $I$ decreases (mass moves closer to the axis), so $\omega$ must increase to keep $L = I\omega$ constant. Quantitatively, if $I$ halves, $\omega$ doubles. This redistribution of rotational inertia is the mechanism behind many spectacular demonstrations of angular momentum conservation in sports, dance, and astronomy.

An important subtlety: while angular momentum is conserved when $\tau_{net}=0$, rotational kinetic energy $K_{rot} = \frac{L^2}{2I}$ is generally not conserved during a change in $I$. When a skater pulls in their arms, they do positive work against the centrifugal tendency of their arms, increasing $K_{rot}$. Specifically, $K_{rot} = \frac{(I\omega)^2}{2I} = \frac{L^2}{2I}$, so as $I$ decreases with $L$ fixed, $K_{rot}$ increases. Energy is conserved overall — the extra kinetic energy comes from the internal work done by the skater's muscles.

• Moment of Inertia — Moment of inertia $I = \sum m_i r_i^2$ is the rotational analogue of mass and appears directly in $L = I\omega$. Understanding how $I$ depends on mass distribution is essential for applying angular momentum conservation.
• Torque — Torque is the rotational analogue of force and is defined by $\vec{\tau} = \vec{r} \times \vec{F}$. The condition for angular momentum conservation is precisely that the net external torque equals zero, making torque conceptually central to this topic.
• Conservation of Linear Momentum — Linear momentum conservation ($\vec{F}_{net}=0 \Rightarrow \vec{p}=$ const) is the direct linear analogue. Comparing the two conservation laws deepens understanding of both and highlights the parallel structure of translational and rotational dynamics.
• Rotational Kinetic Energy — Rotational KE ($K = \frac{1}{2}I\omega^2 = \frac{L^2}{2I}$) is related to angular momentum but is not generally conserved when $I$ changes. Distinguishing between conservation of $L$ and conservation of $K_{rot}$ is a key conceptual challenge in this topic.
• Orbital Mechanics and Kepler's Laws — Kepler's second law — equal areas swept in equal times — is a direct consequence of angular momentum conservation for planets orbiting the Sun under a central (radial) gravitational force that exerts zero torque about the Sun.