Simple Harmonic Motion, Waves, and Optics Study Guide

Simple Harmonic Motion (SHM) and Energy in Vibrating Systems

• Definition of Motion: When the position of an object changes as a function of time, it is in motion. Types include translational, circular, rotational, and oscillatory/vibrational motion.
• Newton's Second Law: The core governing equation is $F_{net} = m\,a$ (1), where $F_{net}$ is the net force, $m$ is the mass, and $a$ is acceleration. For any type of motion, this equation can be solved to find position as a function of time, as it is a second-order differential equation.
• Oscillatory Motion: This is back-and-forth motion over the same path (linear or circular). Types include forced, free, damped, and simple harmonic motion.
• Simple Harmonic Motion (SHM): The simplest oscillatory motion. Characteristics:
  • Acceleration is directed toward a fixed point (equilibrium).
  • Restoring force ($F_r$) is proportional to the displacement from equilibrium.
  • Equation of motion: $F_r = m\,a = m\,\frac{d^2x}{dt^2}$ (2).
• Examples of SHM:
  • Simple pendulum.
  • Mass-spring system on a frictionless surface.
  • LC circuit.
  • Hanged mass-spring system.
• Mass-Spring System Equilibrium: The position where the spring exerts no force. Compression (moving left) or stretching (moving right) creates a restoring force to return the mass to equilibrium.

Mathematical Solutions and Descriptive Terms of SHM

• Differential Equation: From Newton's law, $F = m\,a = m\,\frac{d^2x}{dt^2} = -k\,x$. Rearranging yields $\ddot{x} + \frac{k}{m}\,x = 0$.
• Solution: The general solution is $x(t) = A\cos(\omega\,t)$ (3).
• Angular Frequency (\omega): Derived by substituting the solution into the differential equation: $m(-A\omega^2\cos(\omega\,t)) = -k(A\cos(\omega\,t))$, which leads to $\omega^2 = \frac{k}{m}$ or $\omega = \sqrt{\frac{k}{m}}$.
• SHM Terms:
  • Displacement (x): Distance from equilibrium at any moment (plus or minus sign).
  • Amplitude (A): Maximum displacement from equilibrium.
  • Cycle: One complete to-and-fro motion (e.g., from $x = A$ to $x = -A$ and back to $x = A$).
  • Period (T): Time for one cycle, measured in seconds.
  • Frequency (f): Cycles per second, measured in Hertz ($Hz$). $f = \frac{1}{T}$ and $T = \frac{1}{f}$ (4).
  • Angular Frequency (\omega): Measured in $radians/second$. $\omega = 2\pi\,f = \frac{2\pi}{T}$ (5).
• Calculated Values for Mass-Spring:
  • $f = \frac{1}{2\pi}\,\sqrt{\frac{k}{m}}$
  • $T = 2\pi\,\sqrt{\frac{m}{k}}$ (6)

Application to the Simple Pendulum

• Restoring Force: For a pendulum, $F_r = -m\,g\sin(\theta)$.
• Small Angle Approximation: For small angles, $\sin(\theta) \approx \theta$. Displacement $x = l\,\theta$.
• Equation of Motion: $m\,l\,\frac{d^2\theta}{dt^2} = -m\,g\,\theta$. Dividing by $m\,l$ gives $\frac{d^2\theta}{dt^2} = -\frac{g}{l}\,\theta$ (8).
• Angular Frequency: $\omega = \sqrt{\frac{g}{l}}$ (9).
• Position as Function of Time: $\theta(t) = \theta_o\cos(\sqrt{\frac{g}{l}}\,t)$ (10).
• Period and Frequency:
  • $f = \frac{1}{2\pi}\,\sqrt{\frac{g}{l}}$ (11)
  • $T = 2\pi\,\sqrt{\frac{l}{g}}$ (12)

Kinematics of Simple Harmonic Oscillators

• Velocity (V): The time derivative of displacement: $V(t) = \frac{dx}{dt} = -\omega\,A\sin(\omega\,t) = -V_{max}\sin(\omega\,t)$ (13).
  • $V^2(t) = \omega^2(A^2 - x^2)$ (14).
  • Maximum velocity ($V_{max} = \omega\,A$) occurs at the equilibrium position ($x = 0$).
• Acceleration (a): The time derivative of velocity: $a(t) = \frac{d^2x}{dt^2} = -\omega^2\,A\cos(\omega\,t) = -\omega^2\,x(t)$ (15).
  • Acceleration is maximum when displacement is maximum ($x = A$).
• Calculation Example: A mass-spring system with $m = 50\,g$ ($0.05\,kg$) and $k = 0.20\,N\,m^{-1}$, amplitude $5\,cm$ ($0.05\,m$).
  • At $t = 1.0\,s$, $x(1.0) = 0.05\cos(\sqrt{\frac{0.2}{0.05}} \times 1.0) = 0.05\cos(2.0) = -0.02\,cm$. Note: $\omega\,t$ is in radians.

Energy in Simple Harmonic Motion

• Potential Energy (PE): Work done to stretch/compress a spring: $PE = \frac{1}{2}\,k\,x^2$ (16).
• Kinetic Energy (KE): Energy of motion: $KE = \frac{1}{2}\,m\,v^2$ (16).
• Total Energy (E): $E = KE + PE = \frac{1}{2}\,m\,v^2 + \frac{1}{2}\,k\,x^2$ (17).
• Energy Conservation:
  • At Extreme Points ($x = \pm A$): $v = 0$, so $E = PE = \frac{1}{2}\,k\,A^2$ (18).
  • At Equilibrium ($x = 0$): All energy is kinetic, $E = KE = \frac{1}{2}\,m\,V_{max}^2$ (19).
  • At Intermediate Points: Energy is partly kinetic and partly potential: $E = \frac{1}{2}\,m\,v^2 + \frac{1}{2}\,k\,x^2 = \frac{1}{2}\,k\,A^2$ (20).

Damped Harmonic Oscillator (Free Oscillation)

• Physical Characteristics: Energy is lost as heat or sound (e.g., a tuning fork). Intensity decreases, but frequency remains constant. The amplitude decays exponentially: $x = A_o\exp(-\beta\,t)\cos(\omega\,t)$ (1).
• Damping Force ($F_d$): Often proportional to velocity: $F_d = -b\,v$ (2), where $b$ is the damping constant and the minus sign indicates operation against motion.
• Equation of Motion: $m\,\frac{d^2x}{dt^2} + b\,\frac{dx}{dt} + k\,x = 0$ (3).
• Standard Parameters:
  • Natural frequency (undamped): $\omega_o^2 = \frac{k}{m}$.
  • Damping parameter: $\gamma = \frac{b}{m}$ (4).
  • General Damped Equation: $\frac{d^2x}{dt^2} + \gamma\,\frac{dx}{dt} + \omega_o^2\,x = 0$ (5).
• Solutions Based on Damping Levels:
  • Light Damping: $\frac{\gamma^2}{4} < \omega_o^2$. Result is oscillatory with decaying amplitude: $x = A_o\exp(-\frac{\gamma}{2}\,t)\cos(\omega\,t + \phi)$ (11). Frequency $\omega = \sqrt{\omega_o^2 - \frac{\gamma^2}{4}}$ (9).
  • Heavy Damping (Overdamping): $\frac{\gamma^2}{4} > \omega_o^2$. No oscillation; system returns sluggishly to equilibrium. Solution: $x = A\exp[\frac{-\gamma}{2} + \sqrt{\frac{\gamma^2}{4} - \omega_o^2}]t + B\exp[\frac{-\gamma}{2} - \sqrt{\frac{\gamma^2}{4} - \omega_o^2}]t$ (16).
  • Critical Damping: $\frac{\gamma^2}{4} = \omega_o^2$. Quickest return to equilibrium without oscillation. Solution: $x = (A + B\,t)\exp(-\frac{\gamma}{2}\,t)$ (19).
• Logarithmic Decrement: A measure of decrease between successive maxima: $\ln(\frac{A_n}{A_{n+1}}) = \frac{\gamma}{2}\,T$.
• Relaxation Time (Modulus of Decay): Time for amplitude to decay to $e^{-1} \approx 0.368$ of its original value. $t = \frac{2m}{b}$.
• Quality Factor (Q): Measures the rate of energy decay: $Q = \frac{\omega'\,m}{b}$ (24). Approximately $Q = \frac{\omega_o\,m}{b} = \frac{\sqrt{k\,m}}{b}$ for small damping (27).
• Energy Dissipation: The rate of energy loss is $\frac{dE}{dt} = -b\,v^2$ (32).

Forced Mechanical Oscillator and Resonance

• Equation for Forced Oscillation: A driving force $F(t) = F_o\cos(\omega_d\,t)$ is applied.
  • $m\,\frac{d^2x}{dt^2} + b\,\frac{dx}{dt} + k\,x = F_o\cos(\omega_d\,t)$ (33).
  • $\frac{d^2x}{dt^2} + \gamma\,\frac{dx}{dt} + \omega_o^2\,x = \frac{F_o}{m}\cos(\omega_d\,t)$ (35).
• Resonance: When $\omega_d = \omega_o$, the response of the system (amplitude) becomes exceptionally large.
• Transient Phenomena: When a driving force is first applied, the system oscillates at both the natural frequency (transient part) and the driving frequency. The transient part decays over time due to damping, leaving only the steady-state oscillation at the driving frequency.

Coupled Oscillations and Normal Modes

• Coupled Oscillators: Systems connected such that energy transfers between them (e.g., loaded strings, molecules).
• Normal Mode: A pattern where all parts of the system vibrate at the same frequency with constant amplitude and phase relationship.
• Two Springs and One Mass: $\ddot{x} = -\frac{k_1 + k_2}{m}\,x$. $\omega = \sqrt{\frac{k_1 + k_2}{m}}$. If springs are identical ($k$), $\omega = \sqrt{\frac{2k}{m}}$.
• Two Masses and Three Springs:
  • Symmetric Mode: Masses move together ($x_1 = x_2$). Center spring is not stretched. $\omega_1 = \sqrt{\frac{k}{m}}$.
  • Antisymmetric Mode: Masses move in opposite directions ($x_1 = -x_2$). Center spring ($k_1$) is stretched. $\omega_2 = \sqrt{\frac{k + 2k_1}{m}}$.
• Coupled Pendula: Two pendula of length $l$ linked by a spring ($k$).
  • Normal Modes:
    • First (Symmetric): Pendula move in phase. $\omega_1 = \sqrt{\frac{g}{l}}$.
    • Second (Antisymmetric): Pendula move out of phase ($180^{\circ}$). $\omega_2 = \sqrt{\frac{g}{l} + \frac{2k}{m}}$.
  • Normal Coordinates: $q_a = x_b + x_a$ and $q_b = x_b - x_a$.

Wave Mechanics: Properties and Behavior

• Sound Waves: Mechanical longitudinal waves requiring a medium.
  • Infrasound: $f < 20\,Hz$.
  • Audible Sound: $20\,Hz - 20\,kHz$.
  • Ultrasound: $f > 20\,kHz$.
• Wave Properties:
  • Velocity: $v = f\,\lambda$ (Note: air $v \approx 343\,m/s$ at $20^{\circ}C$).
  • Intensity: $I = \frac{P}{A}$, measured in decibels ($dB$): $\beta = 10\log_{10}(\frac{I}{I_o})$ where $I_o = 10^{-12}\,W/m^2$.
• Transverse vs. Longitudinal:
  • Transverse: Particles oscillate perpendicular to wave travel (e.g., light, string waves, S-waves).
  • Longitudinal: Particles oscillate parallel to wave travel (e.g., sound, P-waves).
• Wave Phenomena:
  • Reflection: Echoes.
  • Refraction: Bending due to medium change; described by Snell's Law: $\frac{\sin(\theta_i)}{\sin(\theta_r)} = \frac{v_1}{v_2} = \frac{n_2}{n_1}$.
  • Diffraction: Bending around obstacles.
  • Dispersion: Different frequencies travel at different speeds (medium-dependent).
  • Polarization: Confining oscillations to a single direction (transverse waves only).
• Wave Equation in 1-D: $\frac{\partial^2\xi}{\partial x^2} = \frac{1}{u^2}\,\frac{\partial^2\xi}{\partial t^2}$ (4).
  • Wave velocity on a string: $u = \sqrt{\frac{T}{\rho}}$, where $T$ is tension and $\rho$ is linear density.
• Wave Power: Average power transmitted: $P_{ave} = \frac{1}{2}\,\rho\,u\,\omega^2\,A^2$ (16).
• Phase vs. Group Velocity:
  • Phase Velocity ($u_p$): Speed of a point of constant phase: $u_p = \frac{\omega}{k}$.
  • Group Velocity ($u_g$): Speed of the overall envelope/wave packet: $u_g = \frac{d\omega}{dk}$.
  • In a non-dispersive medium, $u_g = u_p$.
• Beats: Interference of two waves with slightly different frequencies: $f_{beat} = |f_1 - f_2|$ (28).
• Doppler Effect: Apparent change in frequency due to relative motion.
  • General Formula: $f_o = f_s(\frac{v \pm v_o}{v \mp v_s})$ (14).
  • Top signs for approaching; bottom signs for departing.

Standing Waves and Resonance in Pipes and Strings

• Standing Waves: Formed by reflection and interference at boundaries.
• Fixed Boundaries (String): Both ends are nodes. $\lambda_n = \frac{2L}{n}$ for $n = 1, 2, 3, ...$
• Open Organ Pipe: Both ends are antinodes. Harmonically same as a string fixed at both ends: $\lambda_n = \frac{2L}{n}$.
• Closed Organ Pipe: One end is a node, the other an antinode. Only odd harmonics exist: $\lambda_n = \frac{4L}{n}$ for $n = 1, 3, 5, ...$

Nature and Propagation of Light

• Dual Nature: Light acts as a particle (photons) and a wave.
• Index of Refraction (n): $n = \frac{c}{v}$ (ratio of speed in vacuum to speed in material). In vacuum, $n = 1$.
• Laws of Geometric Optics:
  • Reflection: $\theta_a = \theta_r$.
  • Refraction (Snell's Law): $n_a\sin(\theta_a) = n_b\sin(\theta_b)$.
  • Into denser material ($n_b > n_a$): Bends toward the normal.
  • Into less dense material ($n_b < n_a$): Bends away from the normal.
• Wavelength in Medium: $\lambda = \frac{\lambda_o}{n}$. Frequency ($f$) does not change.
• Total Internal Reflection (TIR): Occurs when light travels from a denser to a less dense medium ($n_a > n_b$) at an angle greater than the critical angle ($\theta_{crit}$).
  • $\sin(\theta_{crit}) = \frac{n_b}{n_a}$.

Lenses and Spherical Mirrors

• Mirror Equation: $\frac{1}{p} + \frac{1}{q} = \frac{1}{f}$ (12). ($p$ = object distance, $q$ = image distance).
  • Concave Mirror: Focal length $f$ is negative (convention used in some notes) or positive (note specifics on convergence). Reflecting surface curves inward. Focal length $f = \frac{R}{2}$.
  • Convex Mirror: Reflecting surface curves outward. $f$ is positive (behind mirror). Image is virtual, upright, and diminished.
• Lens Makers Equation: $\frac{1}{f} = (n - 1)(\frac{1}{R_1} - \frac{1}{R_2})$ (4).
• Thin Lens Equation: $\frac{1}{f} = \frac{1}{p} + \frac{1}{q}$.
  • Converging (Convex): $f > 0$. Real or virtual images.
  • Diverging (Concave): $f < 0$. Always forms virtual, upright, diminished images.
• Magnification (M): $M = \frac{h'}{h} = -\frac{q}{p}$.
• Aberrations:
  • Spherical: Blurred image due to large angle rays.
  • Chromatic: Different wavelengths focus at different points. Corrected by an achromatic doublet.

The Human Eye and Vision Correction

• Anatomy: Cornea, Iris (controls pupil), Pupil (aperture), Eye Lens (adjustable focal length), Retina (light-sensitive surface at yellow/blind spots), Aqueous/Vitreous humor.
• Accommodation: Ability of ciliary muscles to change lens shape to focus on near or far objects.
• Near Point: Closest distance for clear vision (standard $25\,cm$). Increases with age.
• Far Point: Farthest distance for clear vision (infinity for normal eye).
• Power of a Lens (P): $P = \frac{1}{f}$ ($m^{-1}$, or Diopters $D$). $P = \frac{100}{f(cm)}$.
• Defects:
  • Hyperopia (Farsightedness): Image sits behind retina. Corrected with a convex lens.
  • Myopia (Nearsightedness): Image sits in front of retina. Corrected with a concave lens.

Optical Instruments and Resolution

• Simple Microscope: A single convex lens. Magnifying power $M = \frac{D}{f} + 1$ (image at $D$) or $M = \frac{D}{f}$ (Normal adjustment/image at infinity).
• Compound Microscope: Objective (short $f_o$) and Eyepiece (short $f_e$).
  • $M = M_o \times M_e = (\frac{L}{f_o})(\frac{D}{f_e})$ for normal adjustment (10).
• Telescopes:
  • Refracting: Objective and eyepiece. Angular magnification $M = \frac{f_o}{f_e}$ (Normal adjustment).
  • Reflecting (e.g., Newtonian/Cassegrain): Uses a concave parabolic mirror objective. Advantages: No chromatic aberration, large apertures for faint stars.
• Resolving Power (Rayleigh's Criterion): Ability to distinguish two close point sources.
  • Telescope Limit of Resolution: $\theta = \frac{1.22\lambda}{D}$ (15). RP is $\frac{1}{\theta}$.
  • Microscope Limit of Resolution: $d = \frac{\lambda}{2n\sin(\theta)}$ (16), where $2n\sin(\theta)$ is Numerical Aperture (N.A.).

Examples & Exercises

• Example: Wave Speed: If $y(x,t) = 25\cos(1.6 \times 10^7\,t - 0.25\,x)$, then speed $v_p = \frac{\omega}{k} = \frac{1.6 \times 10^7}{0.25} = 6.4 \times 10^7\,m/s$.
• Example: Critical Damping Impulse: Mass $2.5\,kg$, $k = 600\,N/m$. The damping constant $b$ for critical damping: $b = \sqrt{4\,m\,k} = \sqrt{4(2.5)(600)} = 77.5\,kg/s$.
• Example: Doppler Shift: Train with $120\,Hz$ horn moving at $40\,m/s$ ($v_{sound} = 340\,m/s$).
  • Approaching: $f_o = \frac{340}{340 - 40} \times 120 = 136\,Hz$.
  • Receding: $f_o = \frac{340}{340 + 40} \times 120 = 107\,Hz$.