Physics 142: Lecture 19
z
z
Review: Spring Motion
Simple Harmonic Motion
Waves
ÎWave Motion
ÎWave Properties
ÎStanding Waves and Resonance
z Simple
Harmonic Motion
Îω=2πf =2π/T
Îx(t) = [A] cos(ωt)
Îv(t) = -[Aω] sin(ωt)
Îa(t) = -[Aω2] cos(ωt)
z For
a Spring F = kx
Î amax = (k/m) A
Î Aω2 = (k/m) A
k
m
ω=
Lecture 19, Pg 1
T = 2π
Lecture 19, Pg 2
Example
Example
In Case 1 a mass on a spring oscillates back and forth. In Case
2, the mass is doubled but the spring and the amplitude of the
oscillation are the same as in Case 1.
In which case is the maximum potential energy of the mass and
spring the biggest?
PE = 1/2kx2
KE = 0
PE = 0
KE = KEMAX
same
for both
same
for both
1. Case 1
2. Case 2
3. Same
x=-A
x=0
x=+A
x=-A
x=0
x=+A
maximum potential energy = ½ k A2
Lecture 19, Pg 3
Lecture 19, Pg 4
m
k
Example
More…
In Case 1 a mass on a spring oscillates back and forth. In Case
2, the mass is doubled but the spring and the amplitude of the
oscillation are the same as in Case 1.
In which case is the maximum kinetic energy of the mass the
biggest?
In Case 1 a mass on a spring oscillates back and forth.
In Case 2, the mass is doubled but the spring and the
amplitude of the oscillation are the same as in Case 1.
Which case has the largest maximum velocity?
1. Case 1
2. Case 2
3. Same
1. Case 1
2. Case 2
3. Same
Lecture 19, Pg 5
Lecture 19, Pg 6
Example
Pendulum Motion
If the amplitude of the oscillation (same block and same spring)
is doubled, how would the period of the oscillation change? (The
period is the time it takes to make one complete oscillation)
1. The period of the oscillation would double.
2. The period of the oscillation would be halved
3. The period of the oscillation would stay the same
z For
small angles
Period does not depend on A, or m!
ω=
x
T=
+2A
2π
ω
g
L
= 2π
L
L
g
T
t
x
m
mg
-2A
Lecture 19, Pg 7
Lecture 19, Pg 8
Waves
Example
Suppose a grandfather clock (a simple pendulum) runs
slow. In order to make it run on time you should:
1. Make the pendulum shorter
2. Make the pendulum longer
ω=
g
L
Lecture 19, Pg 9
Waves
v=
λ
T
=λf
Lecture 19, Pg 10
Transverse Waves
(wave speed)
Lecture 19, Pg 11
Lecture 19, Pg 12
Longitudinal Waves
Example
A longitudinal sound wave has a speed of 340 m/s in air. If this wave
produces a tone with a frequency of 1000 Hz, what is its
wavelength?
Solution
Lecture 19, Pg 13
Lecture 19, Pg 14
Principle of Superposition
Example
Light waves travel in a vacuum at a speed of 300,000 km/s . The
frequency of visible light is about 5×1014Hz . What is the
approximate wavelength of the light?
Solution
Lecture 19, Pg 15
Lecture 19, Pg 16
Interference
Interference
Lecture 19, Pg 17
Reflection at a Boundary
z When
a wave travels from one
boundary to another, reflection occurs.
Some of the wave travels backwards
from the boundary
Lecture 19, Pg 18
Standing Waves and Resonance
If, for certain frequencies, there is a standing wave in a
string, such a standing wave is said to be produced at
resonance; and the string is said to resonate at these
certain frequencies , called resonant frequencies.
n = 1:
L = 1⋅
Lecture 19, Pg 19
λ
2
Lecture 19, Pg 20
Standing Waves and Resonance
Standing Waves and Resonance
L = n⋅
n = 2:
L = 2⋅
λ
λ=
λ
2
2
2L
,
n
f =
n = 3:
L = 3⋅
for n = 1, 2, 3, …
v
λ
=n
v
,
2L
for n = 1, 2, 3, …
For n=1, f = v/2L - lowest frequency, called fundamental mode or
the first harmonic. For n=2, we have the second harmonic. For
n=3, we have the third harmonic. Etc.
λ
2
Lecture 19, Pg 21
Lecture 19, Pg 22
Stringed Instruments (Just for Fun)
Stringed Instruments
Guitar (or any stringed instrument):
Guitar (or any stringed instrument):
When you pluck the string you hear (mostly) the fundamental frequency of a
standing wave.
You change the frequency by moving your finger on the fret-board:
v=
τ
µ
v is fixed for a given string,
so f1 = v /2L depends only on L
(i.e. where you put your finger on the neck)
L
(λ = 2L)
v=
τ
µ
same
f1,new = v / 2Lnew
new (higher) fundamental frequency
Lnew
(λnew = 2Lnew)
Lecture 19, Pg 23
Lecture 19, Pg 24
Just for fun...
Just for fun ...
Guitar players know how to get a cool sound by suppressing the fundamental
frequency, leaving higher “harmonic content” in the sound:
When you hit a string, its motion is a superposition of the fundamental mode
as well as a small amount of higher modes:
Lecture 19, Pg 25
Just for fun ...
Guitar players know how to get a cool sound by suppressing the fundamental
frequency, leaving higher “harmonic content” in the sound:
Carefully touching the string in the right spot will kill the fundamental
frequency, but not the 2nd harmonic, since it’s zero there anyway...
Lecture 19, Pg 27
Guitar players know how to get a cool sound by suppressing the fundamental
frequency, leaving higher “harmonic content” in the sound:
Carefully touching the string in the right spot will kill the fundamental
frequency, but not the 2nd harmonic, since it’s zero there anyway...
Lecture 19, Pg 26