11.2 Angular Momentum

Angular momentum is a key concept in physics, describing how objects rotate and resist changes to their rotation. It's a vector quantity, meaning it has both magnitude and direction, and plays a crucial role in understanding the behavior of spinning objects and systems.

From figure skaters to planets, angular momentum helps explain various phenomena. Its conservation is a fundamental principle, leading to fascinating applications in physics and engineering. Understanding angular momentum is essential for grasping rotational motion and its effects on the world around us.

Angular Momentum

Vector nature of angular momentum

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• Angular momentum is a vector quantity represented by the symbol $\vec{L}$ that has both magnitude and direction
• The direction of angular momentum is perpendicular to the plane of rotation and determined by the right-hand rule (thumb points in the direction of $\vec{L}$ when fingers curl in the direction of rotation)
• Angular momentum describes the rotational state of an object or system, relates to the object's resistance to changes in its rotational motion, and its conservation leads to stable rotational motion (gyroscopes, planets orbiting the sun)

Angular momentum in particle systems

• The angular momentum of a particle is calculated using the cross product $\vec{L} = \vec{r} \times \vec{p}$, where $\vec{r}$ is the position vector from the reference point to the particle and $\vec{p}$ is the linear momentum of the particle
• The total angular momentum of a system is the sum of individual particle angular momenta: $\vec{L}_{total} = \sum_{i} \vec{L}_{i}$ (solar system, molecular rotations)
• Torque on a particle is calculated using the cross product $\vec{\tau} = \vec{r} \times \vec{F}$, where $\vec{r}$ is the position vector from the reference point to the particle and $\vec{F}$ is the force acting on the particle
• The total torque on a system is the sum of individual particle torques: [object Object],[object Object] (net torque on a complex machine)

Rigid body angular momentum

• The angular momentum of a rigid body rotating around a fixed axis is calculated using $\vec{L} = I \vec{\omega}$, where $I$ is the moment of inertia of the rigid body and $\vec{\omega}$ is the angular velocity vector
• The moment of inertia depends on the mass distribution and axis of rotation, calculated as [object Object],[object Object] for a point mass and using integration or the parallel axis theorem for extended objects (flywheels, rotating machinery)
• The rotational kinetic energy of a rigid body is given by $KE_{rot} = \frac{1}{2}I\omega^2$, which relates to its angular momentum

Torque effects on rotating bodies

• Torque on a rigid body is calculated using the cross product $\vec{\tau} = \vec{r} \times \vec{F}$, where $\vec{r}$ is the position vector from the axis of rotation to the point of force application and $\vec{F}$ is the force acting on the rigid body
• The relationship between torque and angular acceleration is given by $\vec{\tau} = I \vec{\alpha}$, where $I$ is the moment of inertia of the rigid body and $\vec{\alpha}$ is the angular acceleration vector
• Net torque causes angular acceleration, with clockwise torque leading to counterclockwise angular acceleration and vice versa (opening a door, tightening a bolt)

Conservation of angular momentum

• Conservation of angular momentum states that $\vec{L}_{initial} = \vec{L}_{final}$ when the net external torque is zero
• For a rigid body with a fixed axis, $I_{initial} \omega_{initial} = I_{final} \omega_{final}$, meaning the moment of inertia and angular velocity can change, but their product remains constant
• Applications include:
  1. Spinning figure skaters changing their moment of inertia by extending or retracting their arms to alter rotation rate
  2. Orbiting objects (planets, satellites) maintaining constant angular momentum in the absence of external torques
  3. Angular momentum conservation in collisions and explosions (asteroid impacts, rocket propulsion)
• Precession is a consequence of angular momentum conservation, where a rotating body's axis of rotation slowly changes direction due to an applied torque

Rotational dynamics and forces

• Angular velocity ($\vec{\omega}$) describes the rate of rotation of a body around an axis
• Centripetal force is the force that keeps an object moving in a circular path, acting towards the center of rotation