Angular momentum Knowledge-Based Study

Study notes

1. What is Angular Momentum?

   Angular momentum is the rotational analogue of linear momentum. Just as an object moving in a straight line possesses linear momentum, an object rotating about a fixed point or axis possesses angular momentum.

   Point Particle Formulation

   For a single particle of mass m moving with velocity $\vec{v}$ at position vector $\vec{r}$ relative to a chosen reference point, the angular momentum $\vec{L}$ is defined by the cross product:

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

   The magnitude of $\vec{L}$ is given by:

   $$L = r p \sin\theta = m v r_{\perp}$$

   where $\theta$ is the angle between $\vec{r}$ and $\vec{v}$, and $r_{\perp}$ is the perpendicular distance from the axis of rotation to the line of motion (the lever arm).

   • SI Unit: $\text{kg}\cdot\text{m}^2/\text{s}$ or $\text{J}\cdot\text{s}$
   • Direction: Determined by the right-hand rule, perpendicular to the plane formed by $\vec{r}$ and $\vec{p}$.

   Rigid Bodies About a Fixed Axis

   For an extended rigid object rotating about a fixed, symmetric axis with angular velocity $\omega$, the total angular momentum is the sum of the angular momenta of all individual mass elements composing the object.

   This summation relates directly to the body's moment of inertia $I$:

   $$L = I\omega$$

   Here:

   • $I$ represents the moment of inertia about the axis of rotation, measuring resistance to rotational acceleration.
   • $\omega$ represents the angular velocity in radians per second ($\text{rad/s}$).

   Linear vs. Rotational Dynamics Analogy

   Rotational quantities map directly onto translational concepts:

   • Mass $m$ corresponds to moment of inertia $I$.
   • Linear velocity $v$ corresponds to angular velocity $\omega$.
   • Linear momentum $p = mv$ corresponds to angular momentum $L = I\omega$.

   Newton's Second Law for Rotation

   In translational mechanics, net force equals the time rate of change of linear momentum ($\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}$). Similarly, in rotational dynamics, the net external torque $\vec{\tau}_{\text{net}}$ acting on a system equals the time rate of change of its angular momentum.

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

   When the moment of inertia $I$ remains constant, substituting $L = I\omega$ yields the familiar equation:

   $$\tau_{\text{net}} = I\frac{d\omega}{dt} = I\alpha$$

   where $\alpha$ is the angular acceleration. However, formulating the law in terms of $\frac{d\vec{L}}{dt}$ is more fundamental because it remains valid even when the mass distribution—and therefore the moment of inertia—changes over time.

   Principle of Conservation

   If the net external torque acting on a system is zero ($\vec{\tau}_{\text{ext}} = 0$), its total angular momentum remains constant over time:

   $$\frac{d\vec{L}}{dt} = 0 \implies \vec{L}_{\text{initial}} = \vec{L}_{\text{final}}$$

   For a system whose moment of inertia changes from $I_i$ to $I_f$ without external torque, the angular velocity must adjust proportionally:

   $$I_i \omega_i = I_f \omega_f$$

   Key Physical Examples

   • Figure Skater: When skaters pull their arms inward, their mass concentrates closer to the rotation axis, decreasing $I$. Consequently, their angular velocity $\omega$ increases to conserve $L$.
   • Planetary Orbits (Kepler's Second Law): Because gravitational force acts along the line connecting a planet to the Sun, it produces zero torque about the Sun. Thus, angular momentum is conserved, and planets move faster near perihelion and slower near aphelion.