Angular Momentum and the Cross Product: Principles, Examples, and Applications

Angular Momentum

The Cross Product of Two Vectors

The cross product is a fundamental operation in vector algebra, essential for describing rotational quantities in physics. It produces a vector that is perpendicular to the plane formed by the two input vectors.

• Definition: If u and v are vectors, their cross product is w = u \times v.

• Direction: The resulting vector w is orthogonal to both u and v.

• Right-Hand Rule: The direction of w is given by the right-hand rule: point your index finger along u, your middle finger along v, and your thumb points in the direction of w.

• Anti-commutativity: u \times v = -v \times u.

• Unit Vector Cross Products: Examples include \hat{x} \times \hat{y} = \hat{z}, \hat{y} \times \hat{z} = \hat{x}, \hat{z} \times \hat{x} = \hat{y}.

Cartesian Coordinates: For vectors u = (u_x, u_y, u_z) and v = (v_x, v_y, v_z):

• The cross product is given by: $u \times v = (u_y v_z - u_z v_y,\ u_z v_x - u_x v_z,\ u_x v_y - u_y v_x)$

Cylindrical Coordinates: For vectors u = (u_r, u_\theta, u_z) and v = (v_r, v_\theta, v_z):

• The cross product is: $u \times v = (u_\theta v_z - u_z v_\theta,\ u_z v_r - u_r v_z,\ u_r v_\theta - u_\theta v_r)$

Properties:

• Linearity: $c u \times v = u \times c v = c (u \times v)$ for scalar $c$.

• Distributivity: $w \times (u \pm v) = w \times u \pm w \times v$.

The Moment (Torque)

The moment (or torque) of a force about a point describes the tendency of the force to cause rotation about that point.

• Definition: $\mathcal{M}_O F = OP \times F$

• If $O$ lies on the action line of $F$, then $\mathcal{M}_O F = 0$.

• Both translation and rotation can result from a force, depending on its line of action.

Example: Calculating the Moment

Given $F = F \sin \theta \hat{x} - F \cos \theta \hat{y}$ and $AP = b \hat{x} + h \hat{y}$, the moment about point A is:

• $\mathcal{M}_A F = AP \times F$

• Using the determinant formula: $\mathcal{M}_A F = (0, 0, -b F \cos \theta - h F \sin \theta)$

Quantity of Motion (Linear Momentum)

Linear momentum is a measure of the motion of a particle and is defined as:

• $\vec{p} = m \vec{v}$

• Kinetic energy: $K = \frac{p^2}{2m}$

Newton's Second Law

Newton's second law relates the time derivative of momentum to the net force:

• $\frac{d\vec{p}}{dt} = \sum_{n=1}^N \vec{F}_n = R$

Angular Momentum

Definition and Properties

Angular momentum is the rotational analog of linear momentum, describing the motion of a particle relative to a reference point.

• Definition: $H_O = OP \times \vec{p}$

• For a particle of mass $m$ and velocity $\vec{v}$: $H_O = OP \times m \vec{v}$

• $H_O$ is orthogonal to the plane formed by $OP$ and $\vec{p}$.

Time Derivative and Principle

The time derivative of angular momentum is related to the moment of the net force:

• $\frac{dH_O}{dt} = OP \times R$

• Principle: The moment about point $O$ of all forces acting on $P$ equals the time rate of change of angular momentum of $P$ about $O$.

• Vector equation: $\frac{dH_O}{dt} = \mathcal{M}_O R$

Remarks

• $H_O = 0$ if $OP \parallel \vec{p}$ (no rotation relative to $O$).

• Central forces do not create a moment and thus do not affect angular momentum.

Application: Simple Pendulum

Equation of Motion Using Angular Momentum Principle

The simple pendulum consists of a mass $m$ at point $P$, attached to a string of length $L$ and swinging under gravity. The angular momentum principle is used to derive its equation of motion.

Kinematics

• Position: $OP = L \hat{e}_r$

• Velocity: $\vec{v} = L \dot{\theta} \hat{e}_\theta$

• Momentum: $\vec{p} = m \vec{v} = m L \dot{\theta} \hat{e}_\theta$

• Angular momentum: $H_O = OP \times \vec{p} = m L^2 \dot{\theta} \hat{z}$

• Time derivative: $\frac{dH_O}{dt} = m L^2 \ddot{\theta} \hat{z}$

Forces and Moments

• Tension: $T = -T \hat{e}_r$ (central force, moment is zero)

• Weight: $W = -mg \hat{y} = mg (\cos \theta \hat{e}_r - \sin \theta \hat{e}_\theta)$

• Moment of weight about $O$: $\mathcal{M}_O W = OP \times W = -mgL \sin \theta \hat{z}$

Principle of Angular Momentum and Equation of Motion

• Applying the principle: $\frac{dH_O}{dt} = \mathcal{M}_O T + \mathcal{M}_O W$

• Since $\mathcal{M}_O T = 0$, the equation simplifies to: $m L^2 \ddot{\theta} \hat{z} = -mgL \sin \theta \hat{z}$

• Equation of motion: $\ddot{\theta} + \frac{g}{L} \sin \theta = 0$

Example: The equation above governs the oscillatory motion of the pendulum, and for small angles ($\sin \theta \approx \theta$), it reduces to simple harmonic motion.