Chapter 13Chapter 13

Vibrations and Waves

When x is positive ,F is negative ;

When at equilibrium (x=0), F = 0 ;

When x is negative ,F is positive ;

Hooke’s Law ReviewedHooke’s Law Reviewed

kxF

Sinusoidal OscillationSinusoidal Oscillation

Pen traces a sine wave

Graphing x vs. tGraphing x vs. t

A : amplitude (length, m) T : period (time, s)

A

T

Some VocabularySome Vocabulary

Tt

A

ftA

tAx

2cos

)2cos(

)cos(

Tf

Tf

22

1

f = Frequency = Angular FrequencyT = PeriodA = Amplitude = phase

PhasesPhases

Tt

Ttt

A

Tt

Ax

00 2if

)(2cos

2cos

Phase is related to starting point

90-degrees changes cosine to sine

tt sin2

cos

a

x

v

Velocity is 90 “out of phase” with x: When x is at max,v is at min ....

Acceleration is 180° “out of phase” with x a = F/m = - (k/m) x

Velocity and Velocity and Acceleration vs. Acceleration vs. timetime

vv and and aa vs. t vs. t

taa

tvv

tAx

cos

sin

cos

max

max

Find vmax with E conservation

mk

Av

mvkA

max

2max

2

21

21

Find amax using F=ma

mk

Aa

tmatkA

makx

max

max coscos

What is What is ??Requires calculus. Since

mk

AAv

tAtAdtd

max

sincos

mk

Springs and massesSprings and masses

Formula SummaryFormula Summary

tAa

tAv

tAx

cos

sin

cos

2

mk

Tf

Tf

22

1

Example13.1Example13.1

An block-spring system oscillates with an amplitude of 3.5 cm. If the spring constant is 250 N/m and the block has a mass of 0.50 kg, determine

(a) the mechanical energy of the system

(b) the maximum speed of the block

(c) the maximum acceleration.

a) 0.153 J

b) 0.783 m/s

c) 17.5 m/s2

Example 13.2Example 13.2

A 36-kg block is attached to a spring of constant k=600 N/m. If the block is pulled 3.5 cm away from its equilibrium positions and released from rest at t=0, then at t=0.75 seconds,

a) what is the position of the block?

b) what is the velocity of the block?

a) -3.489 cm

b) -1.138 cm/s

Example 13.3Example 13.3

A 36-kg block is attached to a spring of constant k=600 N/m. If the block is pulled 3.5 cm away from its equilibrium position and is pushed so that is has an initial velocity of 5.0 cm/s at t=0, then at t=0.75 seconds,

a) what is the position of the block?

a) 0.00179 cm

Pendulum DemoPendulum Demo

Simple PendulumSimple Pendulum

xLmg

F

Lx

Lx

x

mgF

22sin

sin

Looks like Hooke’s law (k mg/L)

Simple PendulumSimple Pendulum

mk

Lmgk

xLmg

F

""

/""

)sin(

)cos(

max

max

tLv

tLg

Simple pendulumSimple pendulum

Frequency is independent of mass and amplitude!(for small amplitudes)

Lg

Example 13.4Example 13.4A man enters a tall tower, needing to know its height h. He notes that a long pendulum extends from the roof almost to the ground and that its period is 15.5 s.

(a) How tall is the tower?

(b) If this pendulum is taken to the Moon, where the free-fall acceleration is 1.67 m/s2, what is the period of the pendulum there?

a) 59.7 m

b) 37.6 s

Damped OscillationsDamped Oscillations

In real systems, friction slows motion

Longitudinal (Compression) WavesLongitudinal (Compression) Waves

Sound waves are longitudinal waves

Compression and Transverse Waves Compression and Transverse Waves DemoDemo

Transverse WavesTransverse Waves

Elements move perpendicular to wave motionElements move parallel to wave motion

Snapshot of a Transverse WaveSnapshot of a Transverse Wave

xAy 2cos

wavelength

x

Snapshot of Longitudinal Snapshot of Longitudinal Wave Wave

xAy 2cos

y could refer to pressure or density

Moving WaveMoving Wave

vtx

Ay 2cos

moves to right with velocity v

Fixing x=0,

tv

Ay 2cos

fvv

f ,

Moving Wave: Formula SummaryMoving Wave: Formula Summary

ftx

Ay 2cos

fv

Example 13.5Example 13.5A wave traveling in the positive x direction has a frequency of f = 25.0 Hz as shown in the figure. Find the

a) amplitude

b) wavelength

c) period

d) speed of the wave.

a) 9.0 cm

b) 20.0 cm

c) 0.04 s

d) 500 cm/s

Example 13.6Example 13.6

Consider the following expression for a pressure wave,

where it is assumed that x is in cm,t is in seconds and P will be given in N/m2.

txP 25.575.1cos5.43

a) What is the amplitude?

b) What is the wavelength?

c) What is the period?

d) What is the velocity of the wave?

a) 43.5 N/m2

b) 3.59 cm

c) 1.197 s

d) -3.0 cm/s

Example 13.7Example 13.7

Which of these waves move in the positive x direction?

)4.32.4cos(3.21)

)4.32.4sin(3.21)

)4.32.4cos(3.21)

)4.32.4sin(3.21)

)4.32.4cos(3.21)

txye

txyd

txyc

txyb

txya

Speed of a Wave in a Vibrating Speed of a Wave in a Vibrating StringString

LmT

v

where

For different kinds of waves: (e.g. sound)• Always a square root• Numerator related to restoring force• Denominator is some sort of mass density

Example 13.9Example 13.9

A simple pendulum consists of a ball of mass 5.00 kg hanging from a uniform string of mass 0.060 0 kg and length L. If the period of oscillation for the pendulum is 2.00 s, determine the speed of a transverse wave in the string when the pendulum hangs vertically.

28.5 m/s

Superposition Superposition PrinciplePrinciple

),(),(),( 21 txytxytxy

Traveling waves can pass through each other without being altered.

Reflection – Fixed Reflection – Fixed EndEnd

Reflected wave is inverted

Reflection – Free Reflection – Free EndEnd

Reflected pulse not inverted

Standing Waves (Read 14.8)Standing Waves (Read 14.8)Consider a wave and its reflection:

ftx

Ayy

ftx

ftx

A

ftx

Ay

ftx

ftx

A

ftx

Ay

leftright

left

right

2cos2sin2

2sin2cos2cos2sin

2sin

2sin2cos2cos2sin

2sin

Standing WavesStanding Waves

ftx

Ayy leftright

2cos2sin2

•Factorizes into x-piece and t-piece •Always ZERO at x=0 or x=m/2

ResonancesResonances

F ig 1 4 . 1 6 , p . 4 4 2

S l i d e 1 8

Integral number of half wavelengths in length L

Ln 2

Nodes and anti-Nodes and anti-nodesnodes

A node is a minimum in the pattern An antinode is a maximum

Fundamental, 2nd, 3rd... Fundamental, 2nd, 3rd... HarmonicsHarmonics

Ln 2

F ig 1 4 .1 8 , p . 4 4 3

S lid e 2 5

Fundamental (n=1)

2nd harmonic

3rd harmonic

Example 13.10Example 13.10

A cello string vibrates in its fundamental mode with a frequency of 220 vibrations/s. The vibrating segment is 70.0 cm long and has a mass of 1.20 g.

a) Find the tension in the string

b) Determine the frequency of the string when it vibrates in three segments.

a) 163 N

b) 660 Hz

Loose EndsLoose Ends

45

43

4

L

L

L

4

12 nL

(Organ pipes open at one end)

Example 13.11Example 13.11

An organ pipe of length 1.5 m is open at one end. What are the lowest two harmonic frequencies?

DATA: Speed of sound = 343 m/s

57.2 Hz, 171.5 Hz