1. L 22 – Vibrations and Waves [2]
resonance 
clocks – pendulum 
springs 
harmonic motion 
mechanical waves
sound waves
musical instruments

2. simple harmonic oscillator mass and spring on a frictionless surface
Equilibrium position
frictionless
surface k
spring that can be stretched or compressed A A
k is the spring constant, which measures the
stiffness of the spring in Newtons per meter

3. Some 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

4. follow the mass – position vs. time

5.  Period of the mass-spring system
This formula is good for a mass on a hanging spring or for a horizontal spring on the air track
If the mass is quadrupled, the period is doubled
 Period of a pendulum of length L

6. 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

7. springs  amazing devices!
the harder I pull on a spring,
the harder it pulls back
stretching
the harder I push on
a spring, the harder it
pushes back
compression

8. Springs obey Hooke’s Law
spring force (N)
elastic limit of
the spring
amount of stretching
or compressing in meters
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

9. simple harmonic oscillator
the period of oscillation is longer (takes more time to complete a cycle) if a bigger mass (m) is used
the period gets smaller (takes less time to complete a cycle) if a stronger spring (larger k) is used
Period T = in seconds
the time to complete a full cycle does not depend on where the oscillator is started (period is independent of amplitude)

10. Resonance effects
all systems have certain natural vibration tendencies
the mass/spring system oscillates at a certain frequency determined by its mass, m and the spring stiffness constant, k
When you push a child on a swing you are using resonance to make the child go higher and higher.

11. How resonance works
resonance is a way of pumping energy into a system to make it vibrate
in order to make it work the energy must be pumped in at a rate (frequency) that matches one of the natural frequencies that the system likes to vibrate at.
you pump energy into the child on the swing by pushing once per cycle
The Tacoma Narrows bridge was set into resonance by the wind blowing over it

12. resonance examples mass on spring two tuning forks
shattering the glass

13. Waves
What is a wave? A disturbance that moves through something  rather vague!
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 !

14. Homer trips and creates a longitudinal wave

15. Why are waves important?
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)

16. 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

17. 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

18. 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

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

20. 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

21. SOUND WAVES longitudinal pressure disturbances in a gas
the air molecules jiggle back and forth in the same direction as the wave
the diaphragm of the
speaker moves in and
out

22. Sound – a longitudinal wave

23. The pressure waves make your eardrum vibrate
we can only hear sounds between
30 Hz and 20,000 Hz
below 30 Hz is called infrasound
above 20,000 is called ultrasound

24. I can’t hear you! There is no sound in outer space!
Since sound is a disturbance
in air, without air (that is, in
a vacuum) there is no sound.
There is no sound
in outer space!
vacuum
pump