Ch 08: Periodic & Wave Motion

〰️ Chapter At a Glance

2 s

Seconds Pendulum Period

3×10⁸ m/s

Speed of EM Waves

≈346 m/s (25°C)

Speed of Sound in Air

20–20000 Hz

Human Hearing Range

Periodic Motion — repeats at regular intervals along a fixed path
SHM: acceleration is proportional and opposite to displacement, a = −ω²x
Waves — carriers of energy; two types: Mechanical & Electromagnetic
Sound — longitudinal mechanical wave; travels fastest in solids
Key topics: SHM · Simple Pendulum · EM spectrum · Sound characteristics · Echo · Mach number

📊 Key Formulae — Quick Look

T = 2π/ωTime Period (SHM)

T = 2π√(l/g)Small-angle ideal pendulum; Simple Pendulum Time Period

n = 1/TFrequency | Unit: Hz

v = nλWave Speed

λ = v/nWavelength

v ∝ √TSpeed of Sound ∝ √Temperature

🔄 Periodic Motion & SHM

Periodic Motion

Regular Repetitive Motion

Motion that repeats itself along a fixed path at a regular interval of time.

Motion of a swing
Motion of a fan
Vibrations in musical instruments
Motion of a satellite around Earth

Simple Harmonic Motion (SHM)

Oscillation About Mean Position

In linear SHM, acceleration is proportional and opposite to displacement: a = −ω²x.

Clock's pendulum
Load on a spring
Ball hanging from rigid support

📐 Important Terms in SHM

Term	Meaning	Formula / Unit
Displacement	Change in position of particle relative to origin	metres (m)
Amplitude	Position of maximum displacement	metres (m)
Phase	Phase angle specifying the state within an oscillation cycle	—
Time Period (T)	Time to complete one full vibration	T = 2π/ω (seconds)
Frequency (n)	Number of vibrations per second	n = 1/T \| Hz
Wavelength (λ)	Distance between two successive crests or troughs	λ = v/n \| metres

⚠️ Exam Trap

In SHM, acceleration is NOT constant — it varies with displacement
Acceleration direction = always toward mean position (restoring force)
Frequency (Hz) = 1/Time Period. Higher frequency → shorter time period

🕰️ Simple Pendulum

📘 Definition

A heavy point mass (bob) suspended from a fixed support by a light, inextensible string, free to oscillate under gravity.

T = 2π√(l/g)Small-angle ideal pendulum; Time Period of Simple Pendulum | l = length, g = gravitational acceleration

2 s

Seconds Pendulum Period

∞ (Infinite)

Time Period in Satellite

No Effect

Mass of Bob on T

📋 Conditions Affecting Simple Pendulum

Condition	Effect on Time Period (T)	Reason
Going Up (altitude) on Earth	Increases ↑	g decreases with altitude
Going Down (depth) on Earth	Increases ↑	g decreases with depth
In a Satellite (orbit)	Infinite (∞)	Free fall gives approximately zero effective restoring acceleration in the satellite frame; gravitational field is not zero
Summer (Temperature ↑)	Increases ↑	Length increases with heat
Winter (Temperature ↓)	Decreases ↓	Length decreases with cold
On Moon	Increases ↑	g_moon = g_earth/6 (much lower g)
Change in Mass	No Change	T is independent of mass

⚠️ Exam Trap — Pendulum

T does NOT depend on mass of the bob — most commonly asked trap!
T depends on length (l) and g only
Going up OR down both increase T (g decreases both ways)
In a satellite, pendulum does NOT oscillate — T = ∞
Seconds pendulum: T = 2 s; the quoted 84.6-minute maximum is invalid

〰️ Wave Motion

Wave = Disturbance that carries energy from one place to another
Wave motion = process of transmission of disturbance with a certain velocity

v = λ/T = nλWave Speed | n = frequency, λ = wavelength, T = time period

📂 Types of Waves

Type A

Mechanical Waves — require a medium

LONGITUDINAL WAVES

Particles vibrate along direction of wave
Transmission: compression & rarefaction
Examples: sound in air, seismic P waves, compression waves in a slinky

TRANSVERSE WAVES

Particles vibrate perpendicular to direction of wave
Transmission: crests & troughs
Examples: transverse waves on a stretched string; light is electromagnetic

Type B

Electromagnetic Waves — NO medium required

Generated by high-frequency electrical oscillations
Can travel through vacuum
Nature: Neutral & Transverse
Speed: 3 × 10⁸ m/s (in vacuum)

⚠️ Exam Trap

Sound = Longitudinal wave; Light = Transverse wave
Sound needs a medium; Light does NOT (can travel in vacuum)
EM waves are transverse in nature (NOT longitudinal)

📡 Electromagnetic Wave Spectrum

Radio

Micro

Visible

X-ray

γ-ray

← Increasing Wavelength | Increasing Frequency →

Wave	Wavelength	Discoverer (Year)	Key Use
Radio Wave	> 0.1 m	Hertz (1887 demonstration); Marconi (1895 practical wireless)	Radio & TV communication
Microwave	1 m to 1 mm	Heinrich Hertz (1888)	Satellites, wireless communication
Infrared	1 mm to 700 nm	Herschel (1800)	Remote control, fog photography
Visible Light	700 nm to 400 nm	Newton (1666)	Human vision, electric bulb
Ultraviolet (UV)	400 nm to 1 nm	Ritter (1801)	Forged document detection, food preservation
X-Rays	1 nm to 10⁻³ nm	Roentgen (1895)	Bone fractures, lung diseases
Gamma (γ) Rays	< 10⁻³ nm	Paul Villard (1900)	Cancer treatment, nuclear structure

📘 Special Note

Gamma rays were named by Rutherford, though discovered by Paul Villard (1900). Microwaves are generated by a device called magnetron.

⚠️ Exam Trap — EM Wave Discoverers

Radio → Hertz (1887 demonstration); Marconi (1895 practical wireless) | Microwave → Hertz (1888)
Infrared → Herschel (1800) | UV → Ritter (1801)
X-Ray → Roentgen (1895) | γ-Ray → Paul Villard (1900), named by Rutherford
Visible light spectrum (VIBGYOR): Violet (400 nm) to Red (700 nm)

🔊 Sound Wave — Basics

Sound = hearing sensation in ears; a form of energy
Type: Longitudinal mechanical wave
Requires a medium to travel (cannot travel in vacuum)

📘 Sources of Sound

Clapping · Birds chirping · Lightning · Musical instruments · Vehicle sounds

📊 Characteristics of Sound Waves

1. Loudness

Perceived Strength of Sound

Intensity ∝ (Amplitude)²
Intensity ∝ 1/(Distance)²
Intensity: W/m²; sound level: dB; loudness level: phon. Perceived loudness also depends on frequency.
Noise above 85 dB is harmful for human ears

Sound Source	Loudness (dB)
Normal Breathing	10 dB
Whisper	15–30 dB
General Conversation	30–60 dB
Busy Traffic	70 dB
Jet Plane	140–150 dB

2. Pitch

Shrillness of Sound

Pitch ∝ Frequency
Women's voice → Higher pitch (shrill)
Men's voice → Lower pitch (grave)

3. Quality / Timbre

Distinguishing Quality

Distinguishes two sounds of same pitch & loudness
Tone = single frequency sound
Note = sound of multiple frequencies
Example: Harmonium, sitar, sarangi have different timbre at same frequency

4. Intensity

Sound Energy per Unit Area per Second

Unit: Watt/m²

👂 Audible Range of Sound

< 20 Hz

Infrasonic

Seismic waves, heartbeat

20–20,000 Hz

Audible (Human)

Normal human hearing range

> 20,000 Hz

Ultrasonic

Bat, Dog, Dolphin, Cat

Animal / Entity	Hearing Range
Dolphin	150 – 1,50,000 Hz
Bat	1,000 – 1,20,000 Hz
Dog	15 – 45,000 Hz
Cat	60 – 65,000 Hz
Children	Up to 25,000 Hz
Human (Adult)	20 – 20,000 Hz

📘 Applications of Ultrasonic Waves

Ultrasonography & Echocardiography (medical imaging)
Measuring depth of sea (SONAR)
Detecting submarines, icebergs
Industrial cleaning, welding

⚠️ Exam Trap — Ultrasonic Users

Bats and dolphins use high-frequency echolocation. Dogs and cats can hear ultrasound. SONAR = Sound Navigation And Ranging; many systems use ultrasound.

💨 Speed of Sound in Different Media

📘 Key Rules

Speed of sound: Solid > Liquid > Gas
Speed & wavelength change when medium changes
Frequency remains unchanged when moving between media
Depends on elasticity and density of medium

State	Medium	Speed at 25°C
Solid	Aluminium	6420 m/s
Solid	Nickel	6040 m/s
Solid	Steel	5960 m/s
Solid	Iron	5950 m/s
Solid	Brass	4700 m/s
Solid	Glass	3980 m/s
Liquid	Sea Water	1531 m/s
Liquid	Distilled Water	1498 m/s
Liquid	Ethanol	1207 m/s
Liquid	Methanol	1103 m/s
Gas	Hydrogen	1284 m/s
Gas	Helium	965 m/s
Gas	Air	≈346 m/s (25°C)
Gas	Oxygen	316 m/s
Gas	Sulphur Dioxide	213 m/s

🌡️ Effect of Temperature on Speed of Sound

v ∝ √TSpeed of sound increases with temperature (T in Kelvin)

At constant temperature, pressure has NO effect on speed of sound
Moisture (humidity) ↑ → Speed ↑ (moist air is lighter than dry air)
Rainy days: Sound of whistle heard from farther away (due to moisture)

Temperature	Speed in Air
0°C	≈331 m/s
22°C	344 m/s
25°C	346 m/s

⚠️ Exam Trap — Speed of Sound

Pressure change at constant temperature has NO effect on speed of sound
Speed of sound in air at 25°C is approximately 346 m/s
Sound travels fastest in solids (Aluminium = 6420 m/s)
Frequency does NOT change when medium changes — only speed & wavelength change

🔁 Reflection of Sound, Echo & Reverberation

Reflection of Sound

Sound striking a surface and returning

The phenomenon of sound striking a surface and returning back along the same path.

Echo

Re-hearing of sound

Persistence of sound in human ears = 0.1 seconds
Minimum distance between source and obstacle = 17.2 m (v = 344 m/s, delay = 0.10 s)

Reverberation

Persistence of sound after source stops

Sound persists even after source has stopped
Used in: Megaphone, Loudspeakers, Stethoscope
Cinema halls: walls made rough or covered with sound-absorbent material to prevent reverberation

⚠️ Exam Trap — Echo vs Reverberation

Echo = distinct repetition of sound (min distance 17.2 m (v = 344 m/s, delay = 0.10 s))
Reverberation = prolonged sound due to multiple reflections
Persistence of hearing = 0.1 sec (1/10th of a second)
Stethoscope works on the principle of reverberation/multiple reflection

✈️ Mach Number

Mach No. = Speed of body / Speed of sound (same medium)Named after Ernst Mach

Mach < 1

Subsonic

1 < Mach ≤ 5

Supersonic

Mach > 5

Hypersonic

📘 Examples

Fighter jets (e.g., MiG, F-22) → Supersonic
Space shuttles during re-entry → Hypersonic
Commercial aircraft → Subsonic (~Mach 0.8)

⚠️ Exam Trap — Mach Classification

Supersonic broadly means Mach > 1; hypersonic commonly means Mach > 5
Hypersonic = Mach > 5
Sonic booms arise from shock waves during supersonic motion, including sustained supersonic flight

🎯 High-Yield Exam Points

SHM: acceleration always toward mean position
Simple Pendulum: T = 2π√(l/g) — independent of mass
Seconds pendulum: T = 2 s; no valid universal 84.6-minute maximum
Ordinary gravity pendulum in a freely falling satellite has no restoring oscillation; T → ∞ is a formal limit, not zero gravitational field
Speed of EM waves in vacuum = 3 × 10⁸ m/s
EM waves: transverse, neutral, no medium required
Radio → Hertz (1887 demonstration); Marconi (1895 practical wireless); X-Ray → Roentgen (1895); γ-Ray → Villard (1900), named by Rutherford
Sound = longitudinal mechanical wave
Speed of sound: Solid > Liquid > Gas
Speed of sound in air at 25°C ≈ 346 m/s
Intensity ∝ (Amplitude)² | Pitch ∝ Frequency
Noise > 85 dB is harmful for ears
Human hearing range: 20 Hz to 20,000 Hz
Ultrasonic users: Bat, Dog, Cat, Dolphin
Echo: persistence of hearing = 0.1 sec; min distance = 17.2 m (v = 344 m/s, delay = 0.10 s)
Mach: Subsonic <1 | Supersonic 1–5 | Hypersonic >5
Pressure at constant temp has NO effect on speed of sound
Moisture ↑ → Speed of sound ↑

📋 Formula Quick Reference

T = 2π/ωTime Period (SHM)

T = 2π√(l/g)Small-angle ideal pendulum; Simple Pendulum

n = 1/T | λ = v/nFrequency & Wavelength

v = nλWave Speed

v ∝ √TSound Speed vs Temperature

Mach = v_body / v_soundMach Number

⚠️ Most Common Exam Traps

T of pendulum does NOT depend on mass
Going up AND down both increase pendulum T
Sound = Longitudinal; Light = Transverse
Frequency does NOT change when sound changes medium
Pressure change has NO effect on speed of sound (at const. temp)
γ-Ray named by Rutherford, discovered by Villard
SONAR uses ultrasonic waves (ultrasound is sound above 20 kHz)
Echo min distance = 17.2 m (v = 344 m/s, delay = 0.10 s) (distance depends on sound speed and separation-time criterion)
Hypersonic = Mach > 5 (not Mach > 3)

〰️ Chapter 08: Periodic & Wave Motion