Chapter 13: Vibrations and Waves

Hooke’s Law

• A simple example of vibration motion: an object attached to a spring.

Hooke’s law and oscillation

kxFs

The negative sign means that the forceexerted by the spring is always directedopposite the displacement of the object(restoring force).

A restoring force always pushes or pullsthe object toward the equilibrium position.

No friction

Suggested homework problems:12,33,47,54,58

Hooke’s Law

• A simple example of vibration motion: an object attached to a spring.

Hooke’s law and oscillation (cont’d) No friction

Suppose the object is initially pulleda distance A to the right and releasedfrom rest. Then the object does simpleharmonic motion.

Simple harmonic motion occurs when thenet force along the direction of motionobeys Hooke’s law – when the net forceis proportional to the displacement fromequilibrium point and is always directed toward the equilibriumpoint.

periodic motion

Hooke’s Law

• A simple example of vibration motion: an object attached to a spring.

Hooke’s law and oscillation (cont’d) No friction

Terminology: - The amplitude A is the maximum distance of the object from its equi- librium position. –A<= x <= A. - The period T is the time it takes the object to move through one complete cycle of motion from x=A to x=-A and then back to x=A. - The frequency f is the number of complete cycles or vibrations per unit time. f=1/T.

periodic motion

Hooke’s Law

• Example 13.1 : Measuring the spring constant

Hooke’s law and oscillation (cont’d)

0 kdmgFFF sg

d

mgk

• Harmonic oscillator equation

kxFma

xm

ka

Elastic Potential Energy

• The energy stored in a stretched or compressed spring

Elastic potential energy

2

2

1kxPEs

• The conservation of energy:

fsgisg PEPEKEPEPEKE )()(

isgfsg

nc

PEPEKEPEPEKE

W

)()(

If there are only conservative forces

If there are also non-conservative forces

Elastic Potential Energy

• Conservation of energy :

Elastic potential energy

222

2

1

2

1

2

1kxmvkA

22 xAm

kv

- The velocity is zero at x=+A,-A.- The velocity is at its maximum at x=0

Comparing Simple Harmonic Motion with

Uniform Circular Motion

• A circular motion and its projection

Uniform circular motion

As the turntable rotates with constant angular speed, the shadow of the ballmoves back and forth with simpleharmonic motion.

A

xA

v

v 22

0

sin

A

xAvv

220

22 xAm

kv

c.f. simple harmonic motion

Comparing Simple Harmonic Motion with

Uniform Circular Motion

• Period of oscillation (T)

Period and Frequency

One period is completed when the ball rotates 360o andmoves a distance v0T.

ATv 20 0

2

v

AT

From conservation of energy, for simple harmonic oscillation ofa spring system with the spring constant k at x=0,

k

m

v

AmvkA

0

20

2

2

1

2

1

m

kT 2

Comparing Simple Harmonic Motion with

Uniform Circular Motion

• Frequency (f) and angular frequency ()

Period and Frequency

Frequency is how many complete rotations/cycles a simple harmonicoscillation or uniform circular motion makes per unit time.

m

k

Tf

2

11 Units : hertz (Hz) = cycles per second

Angular frequency is a frequency measured in terms of angle.

m

kf 2

Position, Velocity, and Acceleration

as a Function of Time

• We can obtain an expression for the position of an object with simple harmonic motion as a function of time.

x vs. time

cosAx

fTt

22

t if constant angular speed

)cos( tAx

)2cos( ftAx

Position, Velocity, and Acceleration

as a Function of Time

• We can obtain an expression for the velocity of an object with simple harmonic motion as a function of time.

v vs. t

220 xAA

vv

if constant angularspeed

)cos( tAx

A

v

Tf

2

1

20

)2sin()sin(

)sin()(cos 02220

ftAtA

tvtAAA

vv

)2sin( ftAv

Position, Velocity, and Acceleration

as a Function of Time

• We can obtain an expression for the position of an object with simple harmonic motion as a function of time.

Period and Frequency

if constant angularspeed

)cos( tAx

)2cos(2 ftAa

xxm

ka

sinusoidal

Motion of a Pendulum

• If a force is a restoring one, from an analogy of a Hooke’s law we can prove that the system under influence the force makes simple harmonic oscillation.

Pendulum

Ft=

1 if sinsinsin

s

L

mg

L

smg

L

smgmgFt

In an analogy to Hooke’s law Ft=-kx,

L

mgk

L

g

m

kf 2

g

LT

The motion of a pendulum is not simpleharmonic in general but it is if the angle is small.

Motion of a Pendulum

• In general case, the argument for a pendulum system of a mass attached to a string can be used to an object of any shape.

Physical pendulum

mgL

IT 2

g

L

mgL

mLT 22

2

For a simple pendulum,

2mLI

I: moment of inertia

Damped Oscillation

• In any real systems, forces of frictions retard the motion induced by restoring forces and the system do not oscillate indefinitely.

Oscillation with friction

The friction reduces the mechanical energy of the system as timepasses, and the motion is said to be damped.

Waves

• The world is full of waves: sound waves, waves on a string, seismic waves, and electromagnetic waves such as light, radio waves, TV signals, x-rays, and -rays.

Examples and sources of waves

• Waves are produced by some sort of vibration:Vibration of vocal cords, guitar strings, etc

Vibration of electrons in an antenna, etc

sound

radio waves

Vibration of water water waves

Types of waves• Transverse waves

The bump (pulse) travels to the right with a definite speed: traveling wave Each segment of the rope that is disturbed moves in a direction perpendicular to the wave motion: transverse wave

Waves Types of waves (cont’d)

• Longitudinal waves The elements of the medium undergo displacements parallel to the direction of wave motion: longitudial wave Their disturbance corresponds to a series of high- and low- pressure regions that may travel through air or through any material medium with a certain speed.

sound wave = longitudinal wave

C = compressionR = rarefaction

Waves Types of waves (cont’d)

• Longitudinal-transverse waves

Frequency, Amplitude, and Wavelength

Frequency, amplitude, and wavelength

• Consider a string with one end connected to a blade vibrating according to simple harmonic oscillation.

Amplitude A: The maximum distance the string moves.Wavelength : The distance between two successive crests

Wave speed v: v=x/t=wavelength/period)

Frequency f: v==fwavelength/period)

Examples• Example 13.8: A traveling waveA wave traveling in the positive x-direction. Find the amplitude,wavelength, speed, and period of the wave if it has a frequencyof 8.00 Hz. x=40.0 cm and y=15.0 cm.

s 0.125s 00.8

11

m/s 3.20

m) Hz)(0.400 00.8(

m 0.400cm 40.0x

m 0.150cm 0.15

fT

fv

yA

Frequency, Amplitude, and Wavelength

Examples (cont’d)

• Example 13.9: Sound and lightA wave has a wavelength of 3.00 m. Calculate the frequency of thewave if it is (a) a sound wave, and (b) a light wave. Take the speedof sound as 343 m/s and that of light as 3.00x108 m/s.(a)

Hz 114m 3.00

m/s 343

v

f

(b)Hz 1000.1

m 3.00

m/s 1000.3 88

c

f

Frequency, Amplitude, and Wavelength

Speed of Waves on Strings

Speed of waves on strings• Two types of speed:

The speed of the physical string that vibrates up and down transverse to the string in the y-direction The rate at which the disturbance propagates along the length of the string in the x-direction: wave speed

• For a fixed wavelength, a string under greater tension F has a greater wave speed because the period of vibration is shorter, and the wave advances one wavelength during one period. A string with greater mass per unit length (linear density) vibrates more slowly, leading to a longer period and a slower wave speed.

F

v Dimension analysis: [F]=ML/T2, []=M/L, F/=L2/T2, [F/=L/T=[v]

Speed of Waves on Strings

Example 13.10

• A uniform string has a mass M of 0.0300 kg and a length L of 6.00 m. Tension is maintained in the string by suspending a block of mass m = 2.00 kg from one end.

(a) Find the speed of the wave.

mgFmgFF 0

m/s 6.62/

LM

mgFv

(b) Find the time it takes the pulse to travel from the wall to the pulley.

s 0799.0m/s 62.6

cm 00.5

v

dt

Interference of Waves

Superposition principle

• Tow traveling waves can meet and pass through each other without being destroyed or even altered.• When two or more raveling waves encounter each other while moving through a medium, the resultant wave is found by adding together the displacements of the individual waves point by point.

Interference

constructive interference (in phase)

destructive interference (out of phase)

Interference of Waves

Example 13.10

• A uniform string has a mass M of 0.0300 kg and a length L of 6.00 m. Tension is maintained in the string by suspending a block of mass m = 2.00 kg from one end.

(a) Find the speed of the wave.

mgFmgFF 0

m/s 6.62/

LM

mgFv

(b) Find the time it takes the pulse to travel from the wall to the pulley.

s 0799.0m/s 62.6

cm 00.5

v

dt

Reflection of Waves

Reflection of waves at a fixed end

Reflected wave is inverted

Reflection of Waves

Reflection of waves at a free end

Reflected wave is not inverted