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 with rotational symmetry.

• Linear momentum:

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

• Newton's second law (linear):

• Rotational analog:

Conservation of Angular Momentum

• Angular momentum is conserved in a system with no external torque.

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

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 .

Examples and Applications

• Flipping a Spinning Wheel: When the direction of a spinning wheel is flipped, the change in angular momentum must be compensated by the rotation of the person holding it, conserving total angular momentum.

• Glancing Inelastic Collision: When two skaters collide and hold hands, their combined system rotates about their center of mass. The final angular velocity can be found using conservation of angular momentum.

Direction of Angular Acceleration

• Angular acceleration can be aligned or anti-aligned with , or at an angle, changing both the magnitude and direction of $\vec{\omega}$.

Simple Harmonic Motion (SHM)

Potential Energy and Force in Springs

• Potential energy:

• Restoring force:

Kinematic Equations for SHM

• Position:

• Velocity:

• Acceleration:

• Angular frequency:

• Period:

• Frequency:

Example: Measuring Mass with a Spring

• By measuring the period of oscillation, the mass can be found:

• This method does not rely 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

Angular Acceleration from Circular Motion

• For a pendulum of length , the restoring force is

• Torque:

• Angular acceleration:

Small-Angle Approximation

• For small , (in radians).

• Thus,

Amplitude, Angular Velocity, and Period

• Amplitude:

• Angular velocity:

• Period:

• These formulas are accurate only for small amplitudes.

Physical Pendulum

• For a rigid body swinging about a pivot,

• Small-angle:

• Period:

Waves

The Wave Equation

• The vertical displacement satisfies

• General solution:

• Wave number:

• Angular frequency:

• Velocity:

Energy in a Traveling Wave

• Kinetic energy for a small element:

• Total energy per wavelength:

Standing Waves with Fixed Boundaries

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

• Allowed wavelengths:

• Frequencies:

• For a string under tension and linear density :

Sound – Displacement from Equilibrium

• Sound waves are longitudinal waves where particle displacement is out of phase with pressure maxima.

• Maximum particle velocity coincides with pressure minimum.

Sound in a Tube with One Open End

• For a tube closed at one end and open at the other, only odd harmonics are present.

Doppler Effect

Sound from a Moving Source

• The observed frequency changes if the source or observer is moving relative to the medium.

• For a stationary source:

• For a moving source or observer, the Doppler effect applies.

Doppler Effect Equations

• Source moving toward receiver:

• Source moving away:

• Receiver moving toward source:

• Receiver moving away:

Example: Ambulance

• If the pitch of the siren is higher than usual, the ambulance is coming closer.

Special Relativity

Relativistic Kinetic Energy and Momentum

• At speeds close to the speed of light, classical expressions for kinetic energy and momentum are no longer accurate.

• Total energy:

• Relativistic kinetic energy:

• Relativistic momentum:

• Where

Higgs Decay Example

• In particle physics, the Lorentz factor is used to transform energies and momenta between reference frames.

• For a decay at rest, the energy and velocity of decay products can be calculated using conservation of energy and momentum.

Length Contraction and Time Dilation

• Moving objects appear shorter in the direction of motion:

• Moving clocks run slower:

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 , a muon with a proper lifetime of can travel m before decaying.