Angular momentum is the rotational equivalent of linear momentum in physics. It's crucial for understanding spinning objects, from figure skaters to planets. This concept helps explain why objects rotate faster when their mass distribution changes.

Conservation of angular momentum is a fundamental principle in physics. It states that when no external torque acts on a system, its total angular momentum remains constant. This principle explains various phenomena, from the motion of celestial bodies to the tricks performed by gymnasts.

Angular Momentum

Angular vs linear momentum

• Angular momentum ($L$) is the rotational equivalent of linear momentum ($p$)
  • Linear momentum calculated using $p = mv$, $m$ is mass and $v$ is velocity (bullet fired from a gun)
  • Angular momentum calculated using $L = I\omega$, $I$ is moment of inertia and $\omega$ is angular velocity (spinning flywheel)
• Both angular and linear momentum are vector quantities
  • Linear momentum points in the same direction as linear velocity (arrow fired from a bow)
  • Angular momentum points perpendicular to the plane of rotation, determined by the right-hand rule (spinning top)
• Conservation principles apply to both angular and linear momentum
  • In the absence of external forces, linear momentum is conserved (colliding billiard balls)
  • In the absence of external torques, angular momentum is conserved (orbiting planets)
• Mass affects linear momentum, while moment of inertia affects angular momentum
  • Moment of inertia depends on mass distribution and shape of the object (hula hoop vs solid disk)

Torque effects on angular momentum

• Torque ($\tau$) is the rotational analogue of force in linear systems
  • Torque causes a change in angular momentum over time: $\tau = \frac{dL}{dt}$ (opening a door)
• The net external torque acting on a system determines the change in angular momentum
  • If net external torque is zero, angular momentum is conserved (spinning gyroscope)
• Torque is a vector quantity, with its direction determined by the right-hand rule
  • The direction of torque is perpendicular to the plane formed by the position vector and the force vector (turning a wrench)
• The magnitude of torque depends on the magnitude of the force and the perpendicular distance from the axis of rotation to the line of action of the force: $\tau = rF\sin\theta$ (pushing a merry-go-round)
• Angular impulse is the rotational analog of linear impulse and represents the change in angular momentum due to torque applied over time

Conservation of Angular Momentum

Conservation of angular momentum

• When no external torque acts on a system, total angular momentum is conserved: $L_{\text{initial}} = L_{\text{final}}$ (spinning figure skater)
• For a system with multiple objects, the total angular momentum is the sum of individual angular momenta: $L_{\text{total}} = L_1 + L_2 + ... + L_n$ (solar system)
• When the moment of inertia changes, angular velocity must change to conserve angular momentum: $I_1\omega_1 = I_2\omega_2$
  1. Figure skater starts spinning with arms extended
  2. Figure skater pulls arms inward, decreasing moment of inertia
  3. Angular velocity increases to conserve angular momentum
• When two objects collide and stick together, the total angular momentum before and after the collision remains constant: $I_1\omega_1 + I_2\omega_2 = (I_1 + I_2)\omega_f$ (two asteroids colliding and merging)
• In the absence of external torques, the angular momentum of a system remains constant, even if internal forces or torques are present (binary star system orbiting each other)

Rotational Dynamics

• Rotational inertia (moment of inertia) is a measure of an object's resistance to rotational acceleration
• Angular displacement represents the angle through which an object rotates about its axis of rotation
• Rotational kinetic energy is the energy associated with an object's rotation about an axis
• Centripetal force is the force that keeps an object moving in a circular path, directed toward the center of rotation