Chapter 01
Introduction to Oscillations
Story-Based Introduction
Why Does a Swing Keep Moving?
Imagine pushing a child on a swing. You let go — and it keeps going. Back and forth, back and forth. This is oscillation: a motion that repeats about a fixed point called the equilibrium. The universe is saturated with this rhythm — from quantum vibrations in atoms to the wobble of massive stars. Physics calls this the most fundamental motion in nature. Before Newton had gravity equations, before Faraday had field lines — there was the pendulum, ticking away, whispering the secrets of periodic motion.
All oscillations are periodic · Not all periodic motions are oscillations
Definition
Oscillation
To-and-fro motion about a mean (equilibrium) position. The body returns to the same point periodically.
Examples: swing, pendulum, spring-mass, heartbeat
Examples: swing, pendulum, spring-mass, heartbeat
Definition
Periodic Motion
Any motion that repeats in equal intervals of time — not necessarily to-and-fro.
Examples: Earth revolving around Sun, clock hands rotating
Examples: Earth revolving around Sun, clock hands rotating
Oscillations in Nature — The Big Picture
Heartbeat
The cardiac muscle contracts and relaxes ~70 times/min. Your life is literally timed by biological oscillations. ECG is just a recording of these rhythms.
Water Waves
Water molecules oscillate up-down as the wave passes. The wave travels, but the molecules themselves just bob in place — pure oscillatory motion.
Guitar String
When plucked, the string oscillates transversely. Frequency determines pitch. Amplitude determines loudness. The vibration compresses air → you hear sound.
Atomic Vibration
At temperatures above absolute zero, every atom in every solid oscillates about its lattice position. Temperature IS essentially the average kinetic energy of these atomic oscillations.
Earthquakes
Seismic P-waves and S-waves cause ground oscillations. Buildings have natural frequencies — seismic resonance causes catastrophic collapse, as in 1985 Mexico City.
Engineering Structures
Skyscrapers sway, bridges oscillate. The Tacoma Narrows Bridge (1940) collapsed at resonance. Modern designs include Tuned Mass Dampers (TMDs) to counter oscillations.
Curiosity Box — Galileo's Discovery
1602: Galileo was attending Mass in Pisa Cathedral when he noticed a chandelier swinging in the breeze. Using his own pulse as a timer, he discovered something remarkable: the period didn't change even as the swing amplitude decreased. This isochronous (equal-time) property of the pendulum launched the science of oscillations — and eventually led to accurate clocks, enabling navigation of the high seas.
Key Vocabulary at a Glance
| Term | Symbol | Meaning | SI Unit |
|---|---|---|---|
| Amplitude | A | Maximum displacement from equilibrium | metre (m) |
| Time Period | T | Time for one complete oscillation | second (s) |
| Frequency | f | Oscillations per second; f = 1/T | hertz (Hz) |
| Angular freq. | ω | ω = 2πf = 2π/T | rad/s |
| Phase | φ | Stage of oscillation at any instant | radian |
| Equilibrium | x₀ | Mean position — net restoring force = 0 | — |
Module 02
Simple Harmonic Motion
Live Simulation
Spring-Mass Oscillator — Interactive
x =0.0 px
v =0.0 px/s
a =0.0 px/s²
Core Condition
The SHM Restoring Force
F = −kx
Hooke's Law — the foundation of SHM
Why negative? When displaced right (x > 0), force acts left (F < 0). When displaced left (x < 0), force acts right (F > 0). The force always opposes displacement — this restoring nature creates oscillation.
SHM Criterion
Mathematical Condition
a = −ω²x
Acceleration ∝ −displacement
If a ∝ −x, the motion IS SHM. This is the defining criterion. To verify any system performs SHM, show that its acceleration is proportional to displacement and opposite in direction.
Live Graphs
Displacement · Velocity · Acceleration vs Time
Displacement x(t)
Velocity v(t) — leads x by 90°
Acceleration a(t) — anti-phase with x
When displacement is maximum (extreme): velocity = 0, acceleration = maximum (toward mean). When displacement is zero (mean): velocity = maximum, acceleration = 0. Perfect complementarity.
Stable Equilibrium
Ball in a Bowl
Displaced → restoring force brings it back. SHM is possible here. Potential energy has a minimum at equilibrium.
Unstable Equilibrium
Ball on a Hill
Displaced → force amplifies displacement. Runs away. No oscillation. PE has a maximum here.
Neutral Equilibrium
Ball on a Flat Surface
Displaced → no restoring force. Stays wherever placed. PE is constant throughout.
Module 03
Mathematical Representation
Equation 1
Displacement
x(t) = A sin(ωt + φ)
A — Amplitude (max displacement, metres)
ω — Angular frequency (2π/T, rad/s)
φ — Initial phase (position at t=0, radians)
At φ=0: particle starts at mean, moves positive
ω — Angular frequency (2π/T, rad/s)
φ — Initial phase (position at t=0, radians)
At φ=0: particle starts at mean, moves positive
Equation 2
Velocity
v = ω√(A² − x²)
v_max = Aω at x = 0 (mean position)
v = 0 at x = ±A (extremes)
This position-form is NEET's favourite! Derive from energy conservation: ½mv² = ½mω²(A²−x²)
v = 0 at x = ±A (extremes)
This position-form is NEET's favourite! Derive from energy conservation: ½mv² = ½mω²(A²−x²)
Equation 3
Acceleration
a(t) = −Aω² sin(ωt + φ) = −ω²x
a_max = Aω² at extremes (x = ±A) · a = 0 at mean (x = 0) · Always directed toward equilibrium (hence negative sign) · a is anti-phase with x by exactly 180°
Phasor Diagram
Phase Rotation — Interactive
The rotating phasor (length = A) spins at angular velocity ω. Its vertical projection is the displacement x(t). This geometric picture explains why SHM equations use sine and cosine.
Graph Reading Mastery
| Graph | Slope tells you | Area tells you | Max value | Shape |
|---|---|---|---|---|
| x–t | Velocity | — | A | Sine wave |
| v–t | Acceleration | Displacement | Aω | Cosine wave (x leads v by 90°) |
| a–t | Jerk (da/dt) | Velocity change | Aω² | −Sine (anti-phase with x) |
| F–x | Spring constant k | Work done | kA | Straight line through origin |
| a–x | ω² | — | Aω² | Straight line, negative slope |
Module 04
Energy in SHM
Kinetic Energy
KE = ½mω²(A²−x²)
Maximum at x = 0 (mean) = ½mω²A²
Zero at x = ±A (extremes)
Zero at x = ±A (extremes)
Potential Energy
PE = ½mω²x² = ½kx²
Zero at x = 0 (mean)
Maximum at x = ±A (extremes) = ½mω²A²
Maximum at x = ±A (extremes) = ½mω²A²
Total Energy — The Crown Jewel
E = KE + PE = ½mω²A² = ½kA² = constant
Total energy is independent of position! It depends only on amplitude A and angular frequency ω (or spring constant k). This is conservation of mechanical energy in action. No matter where the particle is, E doesn't change — only its form does.
Interactive Energy Tracker
Drag Position — Watch Energy Transform
KINETIC ENERGY 100%
POTENTIAL ENERGY 0%
TOTAL ENERGY 100% — always
Energy at Key Positions
| Position | x | KE | PE | Note |
|---|---|---|---|---|
| Mean | 0 | E (maximum) | 0 | All energy is kinetic |
| Extreme | ±A | 0 | E (maximum) | All energy is potential; v=0 turning point |
| Quarter | A/2 | ¾E | ¼E | Three-quarters kinetic |
| KE = PE | A/√2 | E/2 | E/2 | NEET favourite! Derive: A²−x²=x² → x=A/√2 |
JEE Advanced Trap
If amplitude doubles (A → 2A): Energy quadruples (E → 4E) because E ∝ A². Similarly, if frequency doubles with same amplitude: E quadruples because E ∝ ω². Most students think E ∝ A — a costly mistake in competitive exams.
Module 05
Spring-Mass System
Live Simulation
Horizontal Spring Oscillator
T = 3.14 s · f = 0.32 Hz · ω = 2.00 rad/s
Derivation
Time Period from First Principles
Step 1: Apply Hooke's Law: F = −kx
Step 2: Apply Newton's 2nd law: ma = −kx → a = −(k/m)x
Step 3: Compare with a = −ω²x → ω² = k/m
Step 4: T = 2π/ω = 2π/√(k/m) = 2π√(m/k)
T = 2π√(m/k)
Period ∝ √m · Period ∝ 1/√k · Period is INDEPENDENT of amplitude!
Series Combination
1/k_eff = 1/k₁ + 1/k₂
k_eff = k₁k₂/(k₁+k₂) < min(k₁,k₂)
Weaker system → larger T.
Equal springs (n in series): k_eff = k/n
Weaker system → larger T.
Equal springs (n in series): k_eff = k/n
Parallel Combination
k_eff = k₁ + k₂
k_eff > max(k₁,k₂)
Stiffer system → smaller T.
Equal springs (n in parallel): k_eff = nk
Stiffer system → smaller T.
Equal springs (n in parallel): k_eff = nk
Vertical Spring
Does Gravity Change the Period?
Short answer: NO! Here's the elegant reason:
New equilibrium: Spring stretches by δ = mg/k under gravity
Measure from new equilibrium: Net force = −k(x + δ) + mg = −kx (gravity cancels!)
Result: T = 2π√(m/k) — identical formula. Gravity only shifts the equilibrium position, not the period.
Static deflection: δ = mg/k
Tip: Measure ALL displacements from the new equilibrium, not natural length!
JEE Spring Trap
A spring of spring constant k is cut into n equal pieces. What is k of each piece? Answer: nk (NOT k/n!). Shorter spring = fewer coils = stiffer spring. k ∝ 1/(natural length). Always verify with limiting cases.
Module 06
Simple Pendulum
Live Simulation
Pendulum Dynamics
T = 1.78 s
Full Derivation
From Forces to Formula
Step 1: Forces on bob — Tension T (along string), Weight mg (downward)
Step 2: Tangential restoring force = −mg sinθ (opposes displacement)
Step 3: Small angle approximation: sinθ ≈ θ (valid for θ < 15°)
Step 4: Arc length x = Lθ → θ = x/L. So F = −mg(x/L) = −(mg/L)x
Step 5: k_eff = mg/L. Then T = 2π√(m/k_eff) = 2π√(mL/mg)
T = 2π√(L/g)
Period depends ONLY on length and gravity — not mass, not amplitude!
All Situations — Period Analysis
| Situation | g_eff | Effect on T |
|---|---|---|
| Earth surface | g | Reference (T = 2π√(L/g)) |
| Moon (g_moon = g/6) | g/6 | T × √6 ≈ 2.45T (slower) |
| Lift accelerating upward (a) | g + a | T decreases (faster swing) |
| Lift accelerating downward (a) | g − a | T increases (slower swing) |
| Free fall / weightlessness | 0 | T → ∞ (no oscillation!) |
| Mountain altitude h | g(1−2h/R) | T slightly increases |
| Second's pendulum | g | T = 2 s → L ≈ 1 m |
The Second's Pendulum
A pendulum with T = 2 seconds is called a "second's pendulum". Its length at Earth's surface (g ≈ 9.8 m/s²): L = gT²/4π² = 9.8×4/4π² ≈ 0.993 m ≈ 1 metre. This elegant coincidence is why early metric proposals considered defining the metre as the length of a second's pendulum. Huygens' pendulum clock (1657) used exactly this concept and was accurate to 15 seconds per day — revolutionary precision for the era.
Module 07
Comparison Tables
SHM vs Uniform Circular Motion
| Property | SHM | UCM |
|---|---|---|
| Path | Linear (straight segment) | Circle |
| Speed | Varies: 0 to Aω | Constant |
| Acceleration magnitude | Varies (0 to Aω²) | Constant = v²/r |
| Restoring force | F = −kx (varies) | Centripetal (constant magnitude) |
| Energy form | KE ↔ PE interchange | Only KE (PE constant) |
| Deep connection | SHM = projection of UCM onto a diameter | |
Spring-Mass vs Simple Pendulum
| Property | Spring-Mass | Simple Pendulum |
|---|---|---|
| Time period formula | T = 2π√(m/k) | T = 2π√(L/g) |
| Depends on mass? | YES | NO |
| Depends on gravity? | NO (horizontal) | YES |
| Depends on amplitude? | No | No (for small angles) |
| Restoring force type | Elastic (F = −kx) | Gravitational (F = −mg sinθ) |
| k_equivalent | k | mg/L |
| Valid range | All amplitudes | θ < 15° (sinθ ≈ θ) |
Phase Relationships — At a Glance
| Quantity | At Mean (x=0) | At Extreme (x=±A) | Phase vs x(t) |
|---|---|---|---|
| Displacement x | 0 | ±A (max) | Reference (0°) |
| Velocity v | Aω (max) | 0 | Leads x by π/2 (90°) |
| Acceleration a | 0 | −Aω² (max) | Anti-phase with x: π (180°) |
| Force F | 0 | kA (max, toward mean) | Same as acceleration |
Types of Oscillations / Damping
| Type | Behavior | Real-World Example |
|---|---|---|
| Free / Undamped | Oscillates forever, constant amplitude | Ideal spring (theoretical) |
| Underdamped | Oscillates with exponentially decaying amplitude | Car suspension, door closer |
| Critically damped | Returns to equilibrium fastest without oscillating | Galvanometer, ideal door damper |
| Overdamped | Returns slowly to equilibrium, no oscillation | Thick hydraulic shock absorber |
| Forced / Resonance | Driven at natural frequency → huge amplitude | Tacoma Bridge collapse, MRI |
Module 08
Circular Motion Connection
Deep Insight
SHM is Born from Circular Motion
The blue dot on the circle moves with uniform circular motion. Its red projection on the vertical axis traces perfect SHM. The sinusoidal trail on the right IS the displacement-time graph of SHM.
Radius of circle = Amplitude A · Angular velocity of rotation = ω · x(t) = A sin(ωt + φ)
Radius of circle = Amplitude A · Angular velocity of rotation = ω · x(t) = A sin(ωt + φ)
Phasor Addition
Two SHMs, Same Frequency
A_net = √(A₁² + A₂² + 2A₁A₂cosΔφ)
Add phasors (rotating vectors) like ordinary vectors. The resultant phasor's length = net amplitude.
Special Cases
Phase Difference Effects
Δφ = 0 (in phase): A_net = A₁ + A₂ (constructive)
Δφ = π (anti-phase): A_net = |A₁ − A₂| (destructive)
Δφ = π/2: A_net = √(A₁² + A₂²)
Δφ = π (anti-phase): A_net = |A₁ − A₂| (destructive)
Δφ = π/2: A_net = √(A₁² + A₂²)
Why Trigonometry and Oscillations Are Inseparable
The unit circle parametrized by angle θ gives coordinates (cosθ, sinθ). When θ rotates uniformly (θ = ωt), each coordinate oscillates sinusoidally. This is why sin and cos appear naturally in SHM equations — they are projections of circular motion. The entire edifice of oscillation theory rests on this single geometric fact: projection of uniform circular motion = simple harmonic motion.
Module 09
Master Formula Sheet
Core Equations
x = A sin(ωt + φ)
Displacement
v = ω√(A² − x²)
Velocity (position form)
a = −ω²x
Acceleration
Maximum Values
| Quantity | Max Value | Where |
|---|---|---|
| x_max | A | Extremes |
| v_max | Aω | Mean (x=0) |
| a_max | Aω² | Extremes (x=±A) |
| F_max | kA = mω²A | Extremes |
| KE_max | ½mω²A² | Mean position |
| PE_max | ½mω²A² | Extreme positions |
Time Periods
T = 2π√(m/k)
Spring-Mass
T = 2π√(L/g)
Simple Pendulum
T = 2π√(LC)
LC Circuit oscillations
T = 2π√(I/C)
Torsional pendulum (I=moment of inertia, C=torsional constant)
Frequency Relations
f = 1/T = ω/2π
All three are equivalent
ω = 2πf = 2π/T
Angular frequency
ω = √(k/m)
Spring-mass system
ω = √(g/L)
Simple pendulum
Dimensional Analysis — Every Quantity
| Quantity | Symbol | Dimension | SI Unit |
|---|---|---|---|
| Spring constant | k | [MT⁻²] | N/m |
| Angular frequency | ω | [T⁻¹] | rad/s |
| Time period | T | [T] | second (s) |
| Amplitude | A | [L] | metre (m) |
| Total energy | E | [ML²T⁻²] | Joule (J) |
| Phase | φ | dimensionless | radian |
| Frequency | f | [T⁻¹] | Hz (s⁻¹) |
MNEMONIC — "FATPA" for the five SHM quantities:
Frequency · Amplitude · Time period · Phase · Angular frequency
Key chain: f = 1/T = ω/2π → ω = 2πf = 2π/T
Frequency · Amplitude · Time period · Phase · Angular frequency
Key chain: f = 1/T = ω/2π → ω = 2πf = 2π/T
Module 10
MCQ Master Bank
Click an option to check your answer. Detailed explanations reveal instantly, showing why wrong options fail too.
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Module 11
Numerical Masterclass
Module 12