Angular Momentum and Rotational Motion: Study Notes

Angular Momentum and Rotational Motion

Angular Momentum of a Particle in Circular Motion

When a particle moves in a circular path, it possesses angular momentum relative to a chosen origin. Angular momentum is a vector quantity that describes the rotational equivalent of linear momentum.

• Definition: The angular momentum L of a particle of mass m moving with velocity v at a position vector r from the origin is given by:

$\vec{L} = \vec{r} \times m\vec{v}$

• Magnitude for Circular Motion: If the particle moves in a circle of radius r with speed v about the origin, the magnitude is:

$L = mvr$

• Direction: The direction of \vec{L} is perpendicular to the plane of motion (using the right-hand rule).

• In terms of Angular Speed: If the angular speed is \omega, then v = \omega r and:

$L = mr^2\omega$

• Example: A particle of mass m moves in a circle of radius r at speed v. Its angular momentum about the center is L = mvr, directed perpendicular to the plane.

Angular Momentum of a Rotating Rigid Body

For a rigid body rotating about a fixed axis, the total angular momentum is the sum of the angular momenta of all its particles.

• Moment of Inertia: The moment of inertia I about the axis is:

$I = \sum m_i r_i^2$

• Angular Momentum: For angular speed \omega:

$L = I\omega$

• Example: A bowling ball of mass 100 g and radius 10 cm spinning at 10 rev/s has angular momentum:

$I = \frac{2}{5} mR^2$ (for a solid sphere) $\omega = 2\pi \times 10$ rad/s $L = I\omega$

Angular Momentum of a System of Particles

When two or more particles are connected (e.g., by a rod), the total angular momentum about a point is the sum of the angular momenta of each particle.

• Example: Two masses (4.00 kg and 3.00 kg) at the ends of a 1.00 m rod rotate in the xy-plane about the center. If each moves at 5.00 m/s, the total angular momentum about the origin is:

$L_{\text{total}} = m_1 v_1 r_1 + m_2 v_2 r_2$

• Where r_1 and r_2 are the distances from the origin to each mass.

Torque and Angular Momentum

Torque is the rotational equivalent of force and is related to the change in angular momentum.

• Definition: The torque \vec{\tau} about a point is:

$\vec{\tau} = \vec{r} \times \vec{F}$

• Relation to Angular Momentum:

$\vec{\tau} = \frac{d\vec{L}}{dt}$

• Example: A particle at \vec{r} = (i + 5j) m with force \vec{F} = (3i + 3j) N. The torque about the origin is:

$\vec{\tau} = \vec{r} \times \vec{F}$

Angular Momentum of a Conical Pendulum

A conical pendulum consists of a mass moving in a horizontal circle while suspended by a string at a constant angle \theta from the vertical.

• Magnitude of Angular Momentum: The angular momentum about the center of the circle is:

$L = \left( m^2 g \ell^2 \sin^4 \theta / \cos \theta \right)^{1/2}$

• Derivation: This result comes from analyzing the forces and geometry of the conical pendulum, relating tension, gravity, and circular motion.

• Application: Used to analyze rotational motion in systems with constraints (e.g., amusement park rides, laboratory setups).

Summary Table: Key Quantities in Rotational Motion

Quantity	Symbol	Formula	SI Unit
Angular Momentum (particle)	L	$L = mvr$	kg·m2/s
Angular Momentum (rigid body)	L	$L = I\omega$	kg·m2/s
Moment of Inertia (point mass)	I	$I = mr^2$	kg·m2
Torque	\tau	$\tau = rF\sin\phi$	N·m
Angular Speed	\omega	$\omega = v/r$	rad/s

Additional info: These notes expand on the homework questions by providing definitions, formulas, and context for angular momentum, torque, and rotational motion, as relevant to a college-level physics course (Ch. 10-11).