The Breathing Universe

NODE 01

What is Oscillation?

• Repeated motion around an equilibrium position.
• Displacement → Restoring tendency.
• Swings, pendulums, springs, vibrating strings.

NODE 02

Equilibrium Position

• Reference point of the oscillator.
• Stable equilibrium: displacement creates restoring force.

NODE 03

Periodic Motion

T, f, ω

• Motion that repeats after a fixed time interval.
• Period (T), Frequency (f = 1/T).
• Angular frequency (ω = 2πf).

NODE 04

Amplitude (A)

• Maximum displacement from equilibrium.
• Magnitude only (peak displacement).
• In ideal SHM, Energy ∝ A².

NODE 05

Phase

• Same f, same A, different position in cycle.
• In phase vs Out of phase.
• Phase angle/constant (φ).

NODE 06

Simple Harmonic Motion

• Restoring acceleration is proportional to displacement.
• Opposite in direction: a = -ω²x.
• Force form: F = -kx.

NODE 07

Dynamic Balance

v=0

|a|=MAX

• At max displacement (A): v=0, |a|=max.
• At equilibrium (0): |v|=max, a=0.

NODE 08

SHM from Circular

• Projection of uniform circular motion onto a diameter.
• Links circular motion → sinusoidal motion → SHM.

NODE 09

Equation of Motion

x = A cos(ωt + φ)

• x = A cos(ωt + φ).
• v = -Aω sin(ωt + φ).
• a = -Aω² cos(ωt + φ) → a = -ω²x.

NODE 10

Mass-Spring System

• F = -kx and F = ma → a = -(k/m)x.
• ω = √(k/m).
• Period: T = 2π√(m/k).

NODE 11

Spring Constant

vs

• The value 'k' describes stiffness.
• Large k = stiffer, Small k = softer.
• Units: N/m (Connects to Realm 08).

NODE 12

Energy of Oscillator

• Total Energy E = ½kA² (constant).
• At any pos: E = ½kx² (U) + ½mv² (K).
• Potential ↔ Kinetic exchange.

NODE 13

Pos-Vel Relation

v vs x

• v² = ω²(A² - x²).
• Closer to equilibrium → faster.
• Closer to extreme → slower.

NODE 14

Accel-Pos Relation

• a = -ω²x.
• Simplest mathematical test for ideal SHM.
• Double displacement → double restoring acceleration.

NODE 15

SHM Graphs

• x-t: Sinusoidal.
• v-t: Sinusoidal, shifted in phase.
• a-t: Opposite in phase to displacement.

NODE 16

Simple Pendulum

• Bob, string, pivot, gravity.
• For small oscillations: T = 2π√(L/g).

NODE 17

Assumptions

X

• Small angular amplitude (sin θ ≈ θ).
• Light/inextensible string, point-like bob.
• Negligible air resistance.

NODE 18

Pendulum Period

• T depends on Length (L) and gravity (g).
• Approximately INDEPENDENT of bob mass.
• Independent of amplitude (for small angles).

NODE 19

Pendulum Energy

• Max displacement: U is max, Speed is zero.
• Equilibrium: U is min, Speed is max.
• Potential ↔ Kinetic.

NODE 20

Physical Pendulum

• A real extended object oscillating about a pivot.
• Involves Moment of Inertia (I), COM, Torque.
• Connects Realm 05 to oscillations.

NODE 21

SHM & Potential En

• Ideal spring: U = ½kx² (parabolic curve).
• Stable equilibrium occurs near a minimum of potential energy.

NODE 22

Energy Exchange

• U max → K increases → K max → U increases → U max...
• Energy doesn't disappear, it changes form.

NODE 23

Coupled Oscillations

• Multiple oscillators connected (springs/pendulums).
• Energy transfer between them. Normal modes.
• Gateway to advanced wave physics.

NODE 24

Damping

• Real oscillators lose energy (friction, air, internal).
• Underdamped: oscillates while decaying.
• Critically / Overdamped: returns without oscillating.

NODE 25

Damped SHM

A(t) = A₀e⁻ᵝᵗ

• Amplitude decreases with time.
• Energy gradually leaves the oscillator.

NODE 26

Forced Oscillation

• Externally driving an oscillator.
• Driving force vs Natural frequency.
• Steady-state amplitude, phase difference.

NODE 27

Natural Frequency

• Characteristic frequency determined by physical properties.
• Spring: f = 1/(2π)√(k/m).

NODE 28

Resonance

• Driving frequency ≈ Natural frequency.
• Response amplitude becomes especially large.
• Maximum energy transfer.

NODE 29

Resonance Curve

• Amplitude vs driving frequency.
• Peak at natural frequency.
• Broadening/reduction due to damping.

NODE 30

Resonance in Real Life

• Musical instruments, bridges, buildings.
• Radio tuning, quartz oscillators.
• Can be useful or dangerous.

NODE 31

Quality Factor (Q)

• Measure of sharpness of resonance & energy loss.
• High Q = narrow resonance, less damping.

NODE 32

Phase & Resonance

• Phase relationship between drive and displacement changes.
• Phase lag, phase lead. Prepares for AC circuits.

NODE 33

Electrical Oscillations

• LC oscillation, electrical resonance.
• Energy exchange: Electric ↔ Magnetic fields.
• Bridge: mechanical ↔ electrical oscillator.

NODE 34

Atomic Oscillations

• Particles in matter vibrate (connects to Realm 12).
• Lattice vibration, Phonons, Temperature.

NODE 35

Sound as Oscillation

• A sound-producing source vibrates (cords, strings).
• Oscillation → disturbance → wave.
• Leads to Realm 14.

NODE 36

Oscillation Lab

• Measure T vs m (spring), T vs L (pendulum).
• Determine k, observe damping/resonance.

NODE 37

Graph Chamber

• T² ∝ L (Pendulum). T² ∝ m (Spring).
• Amplitude vs driving frequency/time.

NODE 38

Misconceptions

X

• "Periodic = SHM" → No, SHM requires a ∝ -x.
• "Amplitude affects period" → Not in ideal SHM.
• "Resonance always destroys" → No, extensively useful.

NODE 39

Solving Engine

ID EQ

F ∝ -x?

ω = √(k/m)

• ID Eq → Disp → Restoring Force → is F ∝ -x?
• If driven: compare driving freq with natural freq.

NODE 40

Master Map

EQ → SHM
↓
RESONANCE

• Eq → Displacement → Restoring F → Oscillation.
• Period → SHM → Phase → Energy Exchange.
• Damping → Forced Oscillation → Resonance → Waves.