Conservation of angular momentum Knowledge-Based Study

Study notes

1. Angular Momentum in Rotational Dynamics

   Angular momentum is the rotational analogue of linear momentum. For a point particle of mass m moving with linear momentum $\vec{p}$ at a position vector $\vec{r}$ relative to a reference origin, the angular momentum $\vec{L}$ is defined by the cross product:

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

   For a rigid body rotating about a fixed, symmetric axis with moment of inertia I and angular velocity $\omega$, the magnitude of angular momentum simplifies to:

   $$L = I\omega$$

   The standard SI unit of angular momentum is kilogram meter squared per second ($\text{kg}\cdot\text{m}^2/\text{s}$), which is equivalent to joule-second ($\text{J}\cdot\text{s}$). Because angular momentum is a vector quantity, its direction along a fixed axis of rotation is determined by the right-hand rule corresponding to the angular velocity vector.

   Newton's Second Law for Rotation

   Linear momentum changes in response to an external net force according to $\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}$. Similarly, rotational dynamics establishes a direct relationship between net external torque and the rate of change of angular momentum.

   Differentiating angular momentum with respect to time yields:

   $$\frac{d\vec{L}}{dt} = \frac{d(\vec{r} \times \vec{p})}{dt} = \left(\frac{d\vec{r}}{dt} \times \vec{p}\right) + \left(\vec{r} \times \frac{d\vec{p}}{dt}\right)$$

   Because $\frac{d\vec{r}}{dt} = \vec{v}$ and $\vec{p} = m\vec{v}$, the term $\vec{v} \times (m\vec{v}) = 0$. Substituting $\frac{d\vec{p}}{dt} = \vec{F}_{\text{net}}$ gives:

   $$\vec{\tau}_{\text{net}} = \frac{d\vec{L}}{dt}$$

   For a system of particles or an extended body, internal forces between constituents act in equal and opposite pairs along the line joining them (Newton's third law in strong form), so their internal torques sum to zero. Therefore, only net external torques can alter the total angular momentum of the system.

   Conservation Law and Conditions

   The principle of conservation of angular momentum states that if the net external torque acting on a system about a chosen axis is zero, the total angular momentum of that system about that axis remains constant over time:

   $$\text{If } \vec{\tau}_{\text{net, ext}} = 0, \text{ then } \frac{d\vec{L}}{dt} = 0 \implies \vec{L}_{\text{initial}} = \vec{L}_{\text{final}}$$

   For a rigid or deformable body rotating about a fixed symmetry axis:

   $$I_i \omega_i = I_f \omega_f$$

   Key implications of this relationship include:

   • Inverse proportionality: If the moment of inertia I decreases, the angular velocity $\omega$ must increase proportionally to keep the product constant.
   • Component-wise conservation: If external torque is zero along one specific axis (for example, the z-axis), the angular momentum component along that axis ($L_z$) is conserved, even if external torques act along perpendicular axes.
   • Deformable systems: Internal forces may reconfigure mass within the system, changing I without affecting the total angular momentum, provided external torques remain zero.

   Rotational Kinetic Energy and Common Phenomena

   While angular momentum is conserved in an isolated system, rotational kinetic energy is not necessarily constant. Rotational kinetic energy can be expressed in terms of angular momentum:

   $$K_{\text{rot}} = \frac{1}{2}I\omega^2 = \frac{L^2}{2I}$$

   When an object reduces its moment of inertia from $I_i$ to $I_f$ (where $I_f < I_i$) under zero external torque:

   • Angular momentum remains identical: $L_f = L_i$.
   • Rotational kinetic energy increases: $K_{\text{rot}, f} > K_{\text{rot}, i}$.
   • The increase in kinetic energy is supplied by the positive mechanical work done by internal forces (such as a figure skater drawing their limbs inward against centrifugal effects).

   Common real-world examples include a spinning figure skater, a diver performing aerial somersaults, the rapid spin of collapsed stellar cores (pulsars), and the varying orbital speed of planets described by Kepler's second law.