L 22 – Vibration and Waves [2]L 22 – Vibration and Waves [2]

resonance resonance clocks – pendulum clocks – pendulum springs springs harmonic motion harmonic motion mechanical wavesmechanical waves sound wavessound waves musical instrumentsmusical instruments

VIBRATING SYSTEMSVIBRATING SYSTEMS

Mass and spring on air trackMass and spring on air track Mass hanging on springMass hanging on spring PendulumPendulum Torsional oscillatorTorsional oscillator

All vibrating systems haveone thing in common restoring force

springs amazing devices!

the harder I pull on a spring, the harder it pulls back

the harder I push ona spring, the harder it

pushes back

stretching

compression

Springs obey Hooke’s Lawsp

rin

g fo

rce

(N

)

amount of stretching or compressing (m)

• the strength of a spring is measured by how much force it provides for a given amount of stretch • we call this quantity k, the spring constant in N/m• magnitude of spring force = k amount of stretch Fspring = k x

elastic limit of the spring

example

• A mass of 2 kg is hung from a spring that has a spring constant k = 100 N/m. By how much will it stretch?

• The downward weight of the mass is balanced by the upward force of the spring.

• w = mg = kx2 kg 10 m/s2 = (100 N/m) x20 N = 100 N/m x x = 0.2 m or 20 cm

X=0

m

x

mg

kx

simple harmonic oscillatormass m and spring on a frictionless surface

0 A

Equilibrium positionk

k is the spring constant, which measures thestiffness of the spring in Newtons per meter

frictionlesssurface

spring that can be stretched or

compressed

A

Terminology• the maximum displacement of an object

from equilibrium is called the AMPLITUDE A

• the time that it takes to complete one full cycle (A B C B A ) is called the PERIOD T of the motion

• if we count the number of full cycles the oscillator completes in a given time, that is called the FREQUENCY f of the oscillator

• frequency f = 1 / period = 1 / T

follow the mass – position vs. time

position

time

+ A

- A

T T T

-A 0 +A

Period of the mass-spring system

2m

Tk

If the mass is quadrupled,

the period is doubled

Period of a pendulum of length L

2L

Tg

Does NOT depend onthe mass

Energy in the simple harmonic oscillator

• a compressed or stretched spring has elastic potential energy

• this elastic potential energy is what drives the system

• if you pull the mass from equilibrium and let go, this elastic PE changes into kinetic energy.

• when the mass passes the equilibrium point, the KE goes back into PE

• if there is no friction the energy keeps sloshing back and forth but it never decreases

Waves moving vibrations• What is a wave? A disturbance that

moves (propagates) through something • Due to the elastic nature of materials• The “wave” - people stand up then sit

down, then the people next to them do the same until the standing and sitting goes all around the stadium.

• the standing and sitting is the disturbance• notice that the people move up and down

but the disturbance goes sideways !

• a mechanical wave is a disturbance that moves through a medium ( e.g. air, water, strings)

• waves carry energy • they provide a means to transport energy from

one place to another• electromagnetic waves (light, x-rays, UV rays,

microwaves, thermal radiation) are disturbances that propagate through the electromagnetic field, even in vacuum (e.g. light from the Sun)

Mechanical waves

• a disturbance that propagates through a medium

• waves on strings• waves in water

– ocean waves– ripples that move outward when a

stone is thrown in a pond

• sound waves – pressure waves in air

transverse wave on a string

• jiggle the end of the string to create a disturbance• the disturbance moves down the string• as it passes, the string moves up and then down• the string motion in vertical but the wave moves in the

horizontal (perpendicular) direction transverse wave• this is a single pulse wave (non-repetitive)• the “wave” in the football stadium is a transverse wave

Wave speed: How fast does it go?

• The speed of the wave moving to the right is not the same as the speed of the string moving up and down. (it could be, but that would be a coincidence!)

• The wave speed is determined by:• the tension in the string more tension higher speed• the mass per unit length of the string (whether it’s a

heavy rope or a light rope) thicker rope lower speed

Harmonic waves – keep jiggling the end of the string up and down

Slinky waves

• you can create a longitudinal wave on a slinky

• instead of jiggling the slinky up and down, you jiggle it in and out

• the coils of the slinky move along the same direction (horizontal) as the wave

SOUND WAVES

• longitudinal pressure disturbances in a gas

• the air molecules jiggle back and forth in the same direction as the wave

• Sound waves cannot propagate in a vacuum DEMO

the diaphragm ofThe speaker moves

in and out

Patm

S N

I can’t hear you!

Since sound is a disturbance in air, without air (that is, ina vacuum) there is no sound.

vacuumpump

Sound – a longitudinal wave

The pressure waves make your eardrum vibrate

• we can only hear sounds between about

30 Hz and 20,000 Hz• below 30 Hz is called

infrasound• above 20,000 is called

ultrasound

Sound and Music

• Sound pressure waves in a solid, liquid or gas

• The speed of sound vs

• Air at 20 C: 343 m/s = 767 mph 1/5 mile/sec• Water at 20 C: 1500 m/s• copper: 5000 m/s

• Depends on density and temperature

5 second rule for thunderand lightning

Why do I sound funny whenI breath helium?

• Sound travels twice as fast in helium, because Helium is lighter than air

• Remember the golden rule vs = • The wavelength of the sound waves you

make with your voice is fixed by the size of your mouth and throat cavity.

• Since is fixed and vs is higher in He, the frequencies of your sounds is twice as high in helium!

Acoustic resonance

• tuning fork resonance• shattering the glass