Oscillations A

bout E

quilibrium

7.1 Periodic Motion

Periodic Motion – repeat, same time, same path

Period (T) – time required for one complete cycle (seconds) or seconds/cycle

Frequency (f) – the number of oscillations per second (s-1 or hertz)

7.2 Simple Harmonic Motion

fT

1

7.2 Simple Harmonic Motion

A form of Periodic Motion

Simple Harmonic Motion

A restoring force is applied proportional to the distance from equilibrium

So Hooke’s Law

kxF

7.2 Simple Harmonic Motion

If a graph of simple harmonic motion is created

And spread out over time

We get a wave pattern

Amplitude – maximum

displacement

7.2 Simple Harmonic Motion

7.3 The Period of a Mass on a Spring

The period of a spring is given by the equation

A larger mass would have greater inertia – longer period

A larger spring constant would produce more acceleration, so a shorter period

The period is independent of amplitude

7.3 The Period of a Mass on a Spring

k

mT 2

7.5 The Pendulum

A simple Pendulum

The potential energy

is

So potential energy

is zero at

equilibrium (like SHM)

7.5 The Pendulum

LLcos

L-Lcos

mgyU )cos( LLmgU

The period of a pendulum is given as

Independent of the mass of the bob

7.5 The Pendulum

g

LT 2

Restoring Force

Forces

Components

A pendulum does not act as a

Simple Harmonic Oscillator,

but at small angles

(<30o) it approximates SHM

7.5 The Pendulum

W

T

mgsin mgcos

7.7 Driven Oscillations and Resonance

7.7 Driven Oscillations and Resonance

Natural Frequency – depends on the variables (m,k or L,g) of the object

Forced Vibrations –

caused by an

external force

7.7 Driven Oscillations and Resonance

Resonant Frequency – the natural vibrating frequency of a system

Resonance – when the external frequency is near the natural frequency and damping is small

Tacoma Narrow Bridge

7.8 Types of Waves

7.8 Types of Waves

Mechanical Waves – travels through a medium

The wave travels through the medium, but the medium undergoes simple harmonic motion

Wave motion

Particle motion

7.8 Types of Waves

Waves transfer energy, not

particles

A single bump of a wave is called a pulse

A wave is formed when a force is applied to one end

Each successive particle is moved by the one next to it

7.8 Types of Waves

Parts of a wave

Transverse wave

– particle

motion

perpenduclar to wave motion

Wavelength () measured in meters

Frequency (f) measured in Hertz (Hz)

Wave Velocity (v) meters/second v f

7.8 Types of Waves

Longitudinal (Compressional) Wave

Particles move

parallel to the

direction of wave motion

Rarefaction – where

particles are spread

out

Compression – particles

are close

7.8 Types of Waves

Earthquakes

S wave – Transverse

P wave – Longitudinal

Surface Waves – can travel along the boundary

Notice the circular motion of the particles

7.9 Reflection and Transmission of Waves

7.9 Reflection and Transmission of Waves

When a wave comes to a

boundary (change in

medium) at least some of

the wave is reflected

The type of reflection depends

on if the boundary is fixed

(hard) - inverted

7.9 Reflection and Transmission of Waves

When a wave comes to a

boundary (change in

medium) at least some of

the wave is reflected

Or movable (soft) – in phase

7.9 Reflection and Transmission of Waves

For two or three dimensional we think in terms of wave fronts

A line drawn perpendicular to the wave front is called a ray

When the waves get far from their source and are nearly straight, they are called plane waves

7.9 Reflection and Transmission of Waves

Law of Reflection – the angle of reflection equals the angle of incidence

Angles are always measured from

the normal

i r

7.10 Characteristics of Sound

7.10 Characteristics of Sound

Sound is a longitudinal wave

Caused by the vibration of a medium

The speed of sound depends on the medium it is in, and the temperature

For air, it is calculated as

15.2735.331 K

s

Tv

7.10 Characteristics of Sound

Loudness – sensation of intensity

Pitch – sensation of frequency

Range of human hearing – 20Hz to 20,000 Hz

ultrasonic – higher than human hearing

dogs hear to 50,000 Hz,

bats to 100,000 Hz

infrasonic – lower than human hearing

7.10 Characteristics of Sound

Often called pressure waves

Vibration produces areas of higher pressure

These changes in pressure are recorded by the ear drum

7.11 Intensity of Sound

7.11 Intensity of Sound

Loudness – sensation

Relative to surrounding and intensity

Intensity – power per unit area

Humans can detect intensities

as low as 10-12 W/m2

The threshold of pain

is 1 W/m2

A

PI

7.11 Intensity of Sound

Sound intensity is usually measured in decibels (dB)

Sound level is given as

I – intensity of the sound

I0 – threshold of hearing (10-12 W/m2)

– sound level in dB

Some common relative intensities

0

log10I

I

Source of Sound Sound Level(dB)

Jet Plane at 30 m 140

Threshold of Pain 120

Loud Rock Concert 120

Siren at 30 m 100

Auto Interior at 90 km/h 75

Busy Street Traffic 70

Conversation at 0.50 m 65

Quiet Radio 40

Whisper 20

Rustle of Leaves 10

Threshold of Hearing 0

7.12 The Ear

7.12 The Ear

Steps in sound transmission

7.13 Sources of Sound: Strings and Air Columns

7.13 Sources of Sound

Vibrations in strings

Fundamental frequency

Next Harmonic

L2L

vf

21

LL

vf 2

12 2 ff

7.13 Sources of Sound

Vibrations in strings

Next Harmonic

Strings produce all harmonics – all whole number multiples of the fundamental frequency

L32

L

vf

323 13 3 ff

7.13 Sources of Sound

Vibrations in an open ended tube (both ends)

Fundamental frequency

Next Harmonic

L2L

vf

21

LL

vf 2

12 2 ff

7.13 Sources of Sound

Vibrations in open ended tubes

Next Harmonic

Open ended tubes produce all harmonics – all whole number multiples of the fundamental frequency

Examples include organ pipes and flutes.

L32

L

vf

323 13 3 ff

7.13 Sources of Sound

Vibrations in an closed end tube (one end)

Fundamental frequency

Next Harmonic

L4L

vf

41

L34

L

vf

343 13 3 ff

7.13 Sources of Sound

Vibrations in open ended tubes

Next Harmonic

Closed end tubes produce only odd harmonics

Examples include reeded wind instruments and brass instruments

L54

L

vf

545 15 5 ff

7.14 Interference of Sound Waves: Beats

7.14 Interference of Sound Waves: Beats

If waves are produced by two identical sources

A pattern of constructive and destructive interference forms

Applet

7.15 The Doppler Effect

7.15 The Doppler Effect

Doppler Effect – the change in pitch due to the relative motion between a source of sound and the receiver

Applies to all wave phenomena

Objects moving toward you have a higher apparent frequency

Objects moving away have a lower apparent frequency

Doppler Effect

Light Doppler

7.15 The Doppler Effect

If an object is stationary the equation for the wave velocity is

Sound waves travel outward evenly in all directions

If the object moves toward the observed, the waves travel at the same velocity, but each new vibration is created closer to the observer

fv

Doppler Applet

7.15 The Doppler Effect

The general equation is

The values of Vo (speed of observer) and Vs (speed of source) is positive when they approach each other

s

s VV

VVff 0

Radar Gun

7.16 Interference

7.16 Interference

Interference – two waves pass through the same region of space at the same time

The waves pass through each other

Principle of Superposition – at the point where the waves meet the displacement of the medium is the algebraic sum of their separate displacements

7.16 Interference

Phase – relative position of the wave crests

If the two waves are “in phase”

Constructive interference

If the two waves are “out of phase”

Destructive Interference

7.16 Interference

For a wave (instead of a single phase)

Interference is

calculated by adding

amplitude

In real time this looks

like