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Chapter 13
Vibrations and Waves
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- When x is positive^ , F is negative ;
- When at equilibrium (x=0), F = 0 ;
- When x is negative^ , F is positive ;
Hooke’s Law Reviewed
F =! kx
2
Sinusoidal Oscillation
Pen traces a sine wave
3
Graphing x vs. t
A : amplitude (length, m) T : period (time, s)
A
T
4
Some Vocabulary
f = Frequency ! = Angular Frequency T = Period A = Amplitude " = phase
x = A cos(! t " #)
= A cos( 2 $ ft " #)
= A cos
2 $ t
T
&'^
f =
T
! = 2 " f =
T
5
Phases
Phase is related to starting time
90-degrees changes cosine to sine
x = A cos
2! t
T
%&^
= A cos
2! ( t " t 0 )
T
%&^
()^
if # =
2! t 0
T
cos! t "
%&^
()^
= sin(! t )
6
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 Acceleration vs. time
T
T
T
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vandavs. t
Find vmax with E conservation
Find amax using F=ma
x = A cos! t
v = " v max sin! t
a = " a max cos! t
kA^2 =
mv m^2 ax
v max = A
k
m
! kx = ma
! kA cos " t =! ma max cos " t
a max = A
k
m
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What is!?
Requires calculus. Since
d
dt
A cos! t = "! A sin! t
v max =! A = A
k
m
k m
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Formula Summary
f =
T
! = 2 " f =
T
x = A cos(! t " #)
v = "! A sin(! t " #)
a = "! 2 A (cos! t " #)
k m
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Example13.
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/s^2 11
Example 13.
A 36-kg block is attached to a spring of constant k=600 N/m. The block is pulled 3.5 cm away from its equilibrium positions and released from rest at t=0. 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
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Simple Pendulum
Looks like Hooke’s law (k $ mg/L)
F =! mg sin "
sin " =
x x^2 + L^2
x L
F #!
mg L
x
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Simple Pendulum
F =! mg sin "
sin " =
x x^2 + L^2
x L
F #!
mg L
x
g L
" = "max cos(! t # $)
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Simple pendulum
Frequency independent of mass and amplitude! (for small amplitudes)
g L
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Pendulum Demo
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Example 13.
A 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/s^2 , what is the period of the pendulum there?
a) 59.7 m
b) 37.6 s
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Damped Oscillations
In real systems, friction slows motion
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TRAVELING WAVES
- Sound
- Surface of a liquid
- Vibration of strings
- Electromagnetic
- Radio waves
- Microwaves
- Infrared
- Visible
- Ultraviolet
- X-rays
- Gamma-rays
- Gravity
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Longitudinal (Compression) Waves
Sound waves are longitudinal waves
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Compression and Transverse Waves Demo
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Transverse Waves
Elements move perpendicular to wave motion Elements move parallel to wave motion
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Snapshot of a Transverse Wave
wavelength
x
y = A cos 2!
x
&'^
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Snapshot of Longitudinal Wave
%
y could refer to pressure or density
y = A cos 2!
x
&'^
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Example 13.7b
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/m^2.
What is the wavelength? a) 0.5 cm b) 1 cm c) 1.5 cm d) # cm e) 2# cm
P = 60! cos ( 2 x " 3 t )
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Example 13.7c
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/m^2. What is the frequency? a) 1.5 Hz b) 3 Hz c) 3/# Hz d) 3/(2#) Hz e) 3#/2 Hz
P = 60! cos ( 2 x " 3 t )
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Example 13.7d
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/m^2.
What is the speed of the wave? a) 1.5 cm/s b) 6 cm/s c) 2/3 cm/s d) 3#/2 cm/s e) 2/# cm/s
P = 60! cos ( 2 x " 3 t )
39
Example 13. Which of these waves move in the positive x direction?
a) 5 and 6 b) 1 and 4 c) 5,6,7 and 8 d) 1,4,5 and 8 e) 2,3,6 and 7
1 ) y = !21.3" cos(3.4 x + 2.5 t )
2 ) y = !21.3" cos(3.4 x! 2.5 t )
3 ) y = !21.3" cos(!3.4 x + 2.5 t )
4 ) y = !21.3" cos(!3.4 x! 2.5 t )
5 ) y = 21.3" cos(3.4 x + 2.5 t )
6 ) y = 21.3" cos(3.4 x! 2.5 t )
7 ) y = 21.3" cos(!3.4 x + 2.5 t )
8 ) y = 21.3" cos(!3.4 x! 2.5 t )
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Speed of a Wave in a Vibrating String
For different kinds of waves: (e.g. sound)
- Always a square root
- Numerator related to restoring force
- Denominator is some sort of mass density
v =
T
where μ =
m L
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Example 13.
A string is tied tightly between points A and B as a communication device. If one wants to double the wave speed, one could:
a) Double the tension b) Quadruple the tension c) Use a string with half the mass d) Use a string with double the mass e) Use a string with quadruple the mass
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Superposition Principle
Traveling waves can pass through each other without being altered.
y ( x , t ) = y 1 ( x , t ) + y 2 ( x , t )
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Reflection – Fixed End
Reflected wave is inverted
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Reflection – Free End
Reflected pulse not inverted
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