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Waves Waves and and Harmonic Harmonic MotionMotion
AP PhysicsAP Physics
M. BlachlyM. Blachly
Review: SHO Equation
Consider a SHO with a mass of 14 grams:
Positions are given in mm
( ) 25cos(16 ) 10y t t
SHO Equation
Find the following:
T
k
At t = 3, find
y
Etotal
a
Period of a Harmonic Oscillator
2m
Tk
For a system with a mass m, and a linear restoring force characterized by k, the period is:
Pendulum
For small angles, gravity acts as a linear restoring force.
/
s
mgF x
L
k mg L
x
Pendulum
2L
Tg
Pendulum
A 500 g. mass is suspended from a 2 m long string. It is then pulled to a position of x=4 cm and released. Write the equation for the position of this pendulum at any time.
Damping
The damped Harmonic Oscillator is a system where energy is not conserved.Applet: http://www.lon-capa.org/~mmp/applist/damped/d.htm
In the real world, there is always some damping, from friction, air resistance, internal deformations of the medium, etc.
Damped SHO Examples
Car Spring and shocks
• Lots of damping is good
Radio receivers
• Very little damping is good
Wave Motion
Types of Waves:
• Transverse
• Longitudinal
• Torsional
Characteristics of Waves
• Frequency
• Period
• Amplitude
• Wavelength
• velocity
Wave Speed
v f
Wave Phenomena
Superposition
Interference
Beats
Types of Waves
Longitudinal: The medium oscillates in the same direction as the wave is moving
• Sound
• Slinky demo
Transverse: The medium oscillates perpendicular to the direction the wave is moving.
• Water (more or less)
• Slinky demo
8
Slinky TnR
Suppose that a longitudinal wave moves along a Slinky at a speed of 5 m/s. Does one coil of the slinky move through a distance of five meters in one second?
1. Yes
2. No
5m
12
Harmonic Waves
Wavelength
Wavelength: The distance between identical points on the wave.
Amplitude: The maximum displacement A of a point on the wave.
Amplitude A
A20
y(x,t) = A cos(t –kx)
Angular Frequency: = 2 f
x
y
Wave Number k: k = 2 /
Recall: f = v /
Period and Velocity Period: The time T for a point on the wave to undergo one
complete oscillation.
Speed: The wave moves one wavelength in one period T so its speed is v = / T.
Tv
22
TnR
Suppose a periodic wave moves through some medium. If the period of the wave is increased, what happens to the wavelength of the wave assuming the speed of the wave remains the same?
1. The wavelength increases
2. The wavelength remains the same
3. The wavelength decreases
correct
= v T
26
The wavelength of microwaves generated by a microwave oven is about 3 cm. At what frequency do these waves cause the water molecules in my burrito to vibrate ?
(a) 1 GHz (b) 10 GHz (c) 100 GHz
1 GHz = 109 cycles/sec
The speed of light is c = 3x108 m/s
ACT
29
Recall that v = f.
1 GHz = 109 cycles/sec
The speed of light is c = 3x108 m/s
GHz8
10v 3 10 m sf 10 Hz 10
.03m
H H
O
Makes water molecules rotateMakes water molecules rotate
Solution
30
Absorption coefficientof water as a functionof frequency.
f = 10 GHz
Visible
“water hole”
31
Water and Waves
Interference and Superposition
When too waves overlap, the amplitudes add.
• Constructive: increases amplitude
• Destructive: decreases amplitude
34
ReflectionA slinky is connected to a wall at one end. A pulse travels to the right, hits the wall and is reflected back to the left. The reflected wave is
A) Inverted B) Upright
• Fixed boundary reflected wave is inverted
• Free boundary reflected wave upright
37
Standing Waves Fixed EndpointsFundamental n=1
n = 2L/n
fn = n v / (2L)
Wave Speed
Fv
Standing Waves:
f1 = fundamental frequency (lowest possible)
L / 2
A guitar’s E-string has a length of 65 cm and is stretched to a tension of 82N. If it vibrates with a fundamental frequency of 329.63 Hz, what is the mass of the string?
v = f = 2 (0.65 m) (329.63 s-1) = 428.5 m/s
v2 = F / = F / v2
m= F L / v2
= 82 (0.65) / (428.5)2 = 2.9 x 10-4 kg