Vibration and Waves AP Physics Chapter 11. 11.1 Simple Harmonic Motion Vibration and Waves

Vibration and Waves

AP Physics

Chapter 11

11.1 Simple Harmonic Motion

Vibration and Waves


Periodic motion – when an object vibrates over the same pathway, with each vibration taking the same amount of time

Equilibrium position – the

position of the mass when

no force is exerted on it

11.1


If the spring is stretched from equilibrium, a force acts so the object is pushed back toward equilibrium

Restoring Force

Proportional to the

displacement (x)

Called Hooke’s Law

11.1

F kx


Any vibrating system for which the restoring force is directly proportional to the negative of the displacement (F = -kx) exhibits Simple Harmonic Motion (SHM)


Amplitude (A) – maximum distance from equilibrium

Period (T) – time for one

complete cycle

Frequency (f) – number

of vibrations per

second

11.1

Amplitude

1fT

http://www.ngsir.netfirms.com/englishhtm/SpringSHM.htm


A vertical spring follows the same

pattern

The equilibrium positions is just

shifted by gravity

11.1

11.2 Energy in a Simple Harmonic Oscillator

Vibration and Waves


Review – energy of a spring

So the total mechanical energy of a spring (assuming no energy loss) is

At maximum amplitude then

So E is proportional to the square of the amplitude

11.2

221 kxU s

2212

21 kxmvE

2212

21 )0( kAmE 2

21 kAE


Velocity as a function of position

Since the maximum velocity is when A=0

Factor the top equation for A2,

then combine with the bottom equation

Take square root11.2

2212

21 kxmvE

2212

212

21 kxmvkA 222 kxmvkA 222 xvA k

m )( 222 xAv mk

2212

21 )0(kmvE 2

max21 mvE 2

max212

21 mvkA 2

max2 mvkA 2

max2 vAm

k

)1( 2

222

Ax

mk Av

)1( 2

22max

2

Axvv 2

2

1max Axvv

11.3 The Period and Sinusoidal Nature of SHM

Vibration and Waves


The Period of an object undergoing SHM is independent of the amplitude

Imagine an object traveling in a circular pathway

If we look at the motion in

just the x axis, the

motion is analogous

to SHM

11.3


As the ball moves the displacement in the x changes

The radius is the amplitude

The velocity is tangent to the

circle

Now looking at Components

If we put the angles into the

triangle, we can see

similar triangles11.3

A

Vmax

V22 xA

q

q


So the opposite/hypotenuse is a constant

This can be rewritten

This is the same as the

equation for velocity of

an object in SHM11.3

A

Vmax

V22 xA

q

q

A

xA

v

v 22

max

2

2

max 1 A

xvv


The period would be the time for one complete revolution

The radius is the same as

the amplitude, and the

time for one revolution

is the period11.3

t

xv

t

rv

2

T

Av

2max

max

2

v

AT


Using the previous relationship between maximum velocity and amplitude

Substitute in the top equation

11.3

t

xv

t

rv

2

max

2

v

AT

2max2

1221 mvkA 2

max2 mvkA k

mvA 2max

2

km

vA max

k

mT 2


Position as a function of time

x displacement is

You don’t know this, but

Where is the frequency

So11.3

A

q

cosAx

ft 2

)2cos( ftAx

11.4 The Simple Pendulum

Vibration and Waves


Simple Pendulum – mass

suspended from a cord

Cord is massless (or very

small)

Mass is concentrated in small volume

11.4


Looking at a diagram of a pendulum

Two forces act on the it

1. Weight

2. Tension

The motion of the bob is at a

tangent to the arc

11.4

W

T


The displacement of the bob is given by x

Using the triangle

For a complete circle (360o)

Then for our arc it would be

11.4

W

T

x

qL

Ly2x r

2360

x r

2

360

rx

3602

xL


The component of force in the direction of

motion is

But at small angle

And

So

11.4

W

T

x

qL

Ly

mgsinq

sin sinmg mg F mg

360

2x

L

360

2F mg x

LF kx


Usual standard is below 30o

We can then take the equation and reason that 360o=2p rad (we didn’t study angular motion, so take my word for it)

11.4

360

2F mg x

L

mgF x

Lmg kL


We can now combine the equation for period

Mass does not appear in this equation

The period is independent of mass

11.4

k

mT 22 mg

L

mT 2

LT

g

11.5 Damped Harmonic Motion

Vibration and Waves


The amplitude of a real oscillating object will decrease with time – called damping

Underdamped – takes several swing before coming to rest (above)

11.5


Overdamped – takes a long time to reach equilibrium

Critical damping – equalibrium reached in the shortest time

11.5

11.6 Forced Vibrations; Resonance

Vibration and Waves


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

Forced Vibrations –

caused by an

external force

11.6


Resonant Frequency – the natural vibrating frequency of a system

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

11.6

Tacoma Narrow Bridge

11.7 Wave Motion

Vibration and Waves

11.7 Wave Motion

Mechanical Waves – travels through a medium

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

Wave motion

Particle motion

11.7

11.7 Wave Motion

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

11.7

Tsunami

11.7 Wave Motion

Parts of a wave

Transverse wave

– particle

motion

perpenduclar to wave motion

Wavelength (l) measured in meters

Frequency (f) measured in Hertz (Hz)

Wave Velocity (v) meters/second

11.7

v f

11.7 Wave Motion

Longitudinal (Compressional) Wave

Particles move

parallel to the

direction of wave motion

Rarefaction – where

particles are spread

out

Compression – particles

are close11.7

11.7 Wave Motion

Earthquakes

S wave – Transverse

P wave – Longitudinal

Surface Waves – can travel along the boundary

Notice the circular motion of the particles11.7

11.9 Energy Transported by Waves

Vibration and Waves

11.9 Energy Transported by Waves

Energy for a particle undergoing simple harmonic motion is

Intensity (I) power across a unit area perpendicular to the

direction of energy

flow

So

11.9

221 kAE

22 1

21 2

I r

I r24

PI

r

11.11 Reflection and Transmission of Waves

Vibration and 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

11.11


When a wave comes to a

boundary (change in

medium) at least some of

the wave is reflected

Or movable (soft) – in phase

11.11


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

11.11


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

Angles are always measured from

the normal

11.11

i r

11.12 Interference; Principle of Superposition

Vibration and Waves


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

11.12


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

11.12


For a wave (instead of a single phase)

Interference is

calculated by adding

amplitude

In real time this looks

like

11.12

11.13 Standing Waves; Resonance

Vibration and Waves


In a specific case of interference a standing wave is produced

The areas with complete constructive interference are called loops or antinodes (AN)

The areas with complete destructive interference are called nodes (N)

11.13


Standing waves occur at the natural or resonant frequency of the medium

In this case, called the first harmonic, the wavelength is twice the length of the medium

The frequency of is called the fundamental frequency

11.13

L21


The second harmonic is the next standing wave formed

Then the third harmonic would be

11.13

L2

L32

3


The basic form for the wavelength

of harmonics is

Each resonant frequency, is an integer multiple of the fundamental frequency

Overtone – all the frequencies above the fundamental

The first overtone is the second harmonic

11.13

Lnn2

Harmonic Applet

http://www.physics.smu.edu/~olness/www/05fall1320/applet/pipe-waves.html

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Vibration and Waves AP Physics Chapter 11. 11.1 Simple Harmonic Motion Vibration and Waves