Angular Momentum

Definition and Properties

Angular momentum is the rotational analog of linear momentum and is a fundamental conserved quantity in physics. It is especially important in systems involving rotation or circular motion.

• Linear momentum:

• Angular momentum: , where is the moment of inertia and is the angular velocity.

• Newton's second law (linear):

• Rotational analog: , where is torque and is angular acceleration.

Conservation of Angular Momentum

• Angular momentum is conserved in the absence of external torques.

• Example: Ice Skater – When an ice skater pulls in their arms, their moment of inertia decreases, so their angular velocity increases to keep constant.

• Example: Flipping a Spinning Wheel – If the direction of a spinning wheel is reversed, the person holding it must rotate to conserve total angular momentum.

Angular Momentum as a Cross Product

Angular momentum can also be defined using the cross product:

• The direction of is perpendicular to the plane formed by and (right-hand rule).

• The magnitude is where is the angle between and .

Applications: Collisions and Rotational Systems

• Glancing Inelastic Collision: When two skaters collide and hold hands, their combined system rotates about their center of mass. Angular momentum is conserved about the axis of rotation.

• For each skater:

• After collision:

Direction of Angular Acceleration

• Angular acceleration can be aligned or anti-aligned with , changing the magnitude and/or direction of rotation.

• Torque and angular acceleration are related:

Simple Harmonic Motion (SHM)

Springs and Oscillators

Simple harmonic motion describes systems where the restoring force is proportional to displacement and directed toward equilibrium.

• Potential energy in a spring:

• Restoring force:

• Oscillatory motion occurs when a mass attached to a spring is displaced and released.

Kinematic Equations for SHM

• Position:

• Velocity:

• Acceleration:

• Angular frequency:

• Period:

• Frequency:

Example: Measuring Mass with a Spring

• By measuring the period of oscillation and knowing the spring constant , the mass can be found:

• This method does not depend on gravity and can be used in space.

Vertical Springs

• When a spring is mounted vertically, gravity shifts the equilibrium position.

• Net force:

• Equilibrium shift:

Pendulums and Rotational Oscillators

Angular Acceleration from Circular Motion

• For a pendulum of length , the tangential force is

• Angular acceleration:

Small-Angle Approximation

• For small angles, (in radians).

• Thus,

• This leads to simple harmonic motion for small oscillations.

Amplitude, Angular Velocity, and Period

• Amplitude:

• Angular velocity:

• Period:

• For larger amplitudes, the period deviates from this formula.

Physical Pendulum

• For a rigid body swinging about a pivot, where is the distance from pivot to center of mass.

• Equation of motion:

• For small angles:

• Period:

• For complex shapes, use the parallel-axis theorem to find .

Waves

The Wave Equation

• The vertical displacement of a wave satisfies , where is the wave speed.

• General solution:

• is the wave number (), is angular frequency ().

Energy in a Traveling Wave

• Kinetic energy for a small element:

• Total energy per wavelength:

• Potential and kinetic energy are equal at any instant.

Wave Properties

• Amplitude:

• Angular velocity:

• Wave number:

• Period:

• Wavelength:

• Velocity:

• For a string: where is linear mass density.

Standing Waves with Fixed Boundaries

• Standing waves form when two waves of the same frequency and amplitude travel in opposite directions and interfere.

• For a string of length fixed at both ends: ,

• For a string with mass hanging:

Clicker Question Example

• If mass is quadrupled, frequency doubles and wavelength is unchanged.

Sound and the Doppler Effect

Sound Waves

• Sound waves are longitudinal waves in a medium (air, water, etc.).

• Displacement of particles is out of phase with pressure maxima.

• Maximum particle velocity coincides with pressure minimum.

• No net mass transport occurs; only energy and momentum are transferred.

Sound in a Tube with One Open End

• Musical instruments often use tubes with one end open.

• Standing wave patterns depend on boundary conditions.

• First harmonic: , higher harmonics:

Sound from a Moving Source

• The speed of sound depends only on the medium.

• If the source moves, the wavelength and frequency observed change (Doppler effect).

• Frequency received:

Doppler Effect

• When source and/or observer move, the observed frequency shifts.

• Formulas:

  • Source moving toward receiver:

  • Source moving away:

  • Receiver moving toward source:

  • Receiver moving away:

• Higher pitch (frequency) means the source is approaching.

Clicker Question Example

• If you hear a higher pitch from an ambulance, it is coming closer.

Special Relativity

Relativistic Kinetic Energy and Momentum

• At high speeds, classical kinetic energy is insufficient; use

• Relativistic kinetic energy:

• Total energy:

• Relativistic momentum:

• Where

Higgs Decay Example

• In particle physics, the -factor is used in energy and momentum transformations.

• For decay products, energy and velocity can be calculated using relativistic formulas.

Length Contraction and Time Dilation

• Moving objects appear shorter:

• Moving clocks run slower:

• These effects are only significant at speeds close to the speed of light.

Cosmic Muons

• Muons created in the upper atmosphere live longer (in Earth's frame) due to time dilation, allowing them to reach the surface.

• Example: If , muon lifetime increases from to .

Additional info: These notes cover key topics from college-level physics, including angular momentum, oscillations, waves, sound, and special relativity, with relevant equations, examples, and applications for exam preparation.