Physics 211: Angular Momentum, Oscillations, Waves, Sound, and Special Relativity

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: $\vec{p} = m\vec{v}$

• Angular momentum: $\vec{L} = I\vec{\omega}$, where $I$ is the moment of inertia and $\vec{\omega}$ is the angular velocity.

• Newton's second law (linear): $\vec{F}_{\text{net}} = m\vec{a} = \frac{d\vec{p}}{dt}$

• Rotational analog: $\vec{\tau}_{\text{net}} = I\vec{\alpha} = \frac{d\vec{L}}{dt}$, where $\vec{\tau}$ is torque and $\vec{\alpha}$ 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 $I$ decreases, so their angular velocity $\omega$ increases to keep $L = I\omega$ 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:

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

• The direction of $\vec{L}$ is perpendicular to the plane formed by $\vec{r}$ and $\vec{v}$ (right-hand rule).

• The magnitude is $L = mrv\sin\phi$ where $\phi$ is the angle between $\vec{r}$ and $\vec{v}$.

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: $\vec{L} = \vec{r} \times \vec{p}$

• After collision: $L_f = I_{\text{tot}}\omega_f = 2mr^2\omega_f$

Direction of Angular Acceleration

• Angular acceleration $\vec{\alpha}$ can be aligned or anti-aligned with $\vec{\omega}$, changing the magnitude and/or direction of rotation.

• Torque and angular acceleration are related: $\vec{\tau} = I\vec{\alpha}$

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: $U = \frac{1}{2}kx^2$

• Restoring force: $F = -\frac{dU}{dx} = -kx$

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

Kinematic Equations for SHM

• Position: $x(t) = A\cos(\omega t)$

• Velocity: $v(t) = -A\omega\sin(\omega t)$

• Acceleration: $a(t) = -\omega^2A\cos(\omega t) = -\omega^2x(t)$

• Angular frequency: $\omega = \sqrt{\frac{k}{m}}$

• Period: $T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}$

• Frequency: $f = \frac{1}{T} = \frac{\omega}{2\pi}$

Example: Measuring Mass with a Spring

• By measuring the period $T$ of oscillation and knowing the spring constant $k$, the mass $m$ can be found: $m = \frac{kT^2}{4\pi^2}$

• 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: $F_{\text{net}} = -k(y + y_0) - mg = -k(y_0 + mg/k)$

• Equilibrium shift: $y_0 = -\frac{mg}{k}$

Pendulums and Rotational Oscillators

Angular Acceleration from Circular Motion

• For a pendulum of length $L$, the tangential force is $F_t = mL\alpha = -mg\sin\theta$

• Angular acceleration: $\alpha = -\frac{g}{L}\sin\theta$

Small-Angle Approximation

• For small angles, $\sin\theta \approx \theta$ (in radians).

• Thus, $\alpha = -\frac{g}{L}\theta$

• This leads to simple harmonic motion for small oscillations.

Amplitude, Angular Velocity, and Period

• Amplitude: $A = |\theta_{\text{max}}|$

• Angular velocity: $\omega = \sqrt{\frac{g}{L}}$

• Period: $T = 2\pi\sqrt{\frac{L}{g}}$

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

Physical Pendulum

• For a rigid body swinging about a pivot, $\tau = I\alpha = -mgd\sin\theta$ where $d$ is the distance from pivot to center of mass.

• Equation of motion: $I\frac{d^2\theta}{dt^2} = -mgd\sin\theta$

• For small angles: $\alpha = -\frac{mgd}{I}\theta$

• Period: $T = 2\pi\sqrt{\frac{I}{mgd}}$

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

Waves

The Wave Equation

• The vertical displacement $y(x,t)$ of a wave satisfies $\frac{d^2y}{dt^2} = v^2\frac{d^2y}{dx^2}$, where $v$ is the wave speed.

• General solution: $y(x,t) = A\sin(kx - \omega t + \phi)$

• $k$ is the wave number ($k = \frac{2\pi}{\lambda}$), $\omega$ is angular frequency ($\omega = 2\pi f$).

Energy in a Traveling Wave

• Kinetic energy for a small element: $\Delta K = \frac{1}{2}(\Delta m)v_y^2$

• Total energy per wavelength: $E = K + U = \frac{1}{2}\mu A^2\omega^2\lambda$

• Potential and kinetic energy are equal at any instant.

Wave Properties

• Amplitude: $y_{\text{max}}$

• Angular velocity: $\omega = 2\pi f = \frac{2\pi v}{\lambda}$

• Wave number: $k = \frac{2\pi}{\lambda}$

• Period: $T = \frac{2\pi}{\omega}$

• Wavelength: $\lambda = \frac{v}{f}$

• Velocity: $v = f\lambda$

• For a string: $v = \sqrt{\frac{T_{\text{tension}}}{\mu}}$ where $\mu$ 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 $L$ fixed at both ends: $\lambda_n = \frac{2L}{n}$, $f_n = \frac{nv}{2L}$

• For a string with mass $M$ hanging: $f_n = \frac{n}{2L}\sqrt{\frac{Mg}{\mu}}$

Clicker Question Example

• If mass $M$ 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: $L = \frac{\lambda_1}{4}$, higher harmonics: $L = \frac{(2n-1)\lambda_n}{4}$

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: $f = \frac{v}{\lambda}$

Doppler Effect

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

• Formulas:

  • Source moving toward receiver: $f = f_0 \frac{v}{v - v_s}$

  • Source moving away: $f = f_0 \frac{v}{v + v_s}$

  • Receiver moving toward source: $f = f_0 \frac{v + v_r}{v}$

  • Receiver moving away: $f = f_0 \frac{v - v_r}{v}$

• 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 $E = mc^2 + K$

• Relativistic kinetic energy: $K = (\gamma - 1)mc^2$

• Total energy: $E = \gamma mc^2$

• Relativistic momentum: $p = \gamma mv$

• Where $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$

Higgs Decay Example

• In particle physics, the $\gamma$-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: $L' = L/\gamma$

• Moving clocks run slower: $\Delta t' = \gamma \Delta t$

• 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 $\gamma = 100$, muon lifetime increases from $2.2\ \mu s$ to $220\ \mu s$.

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.