Chapter 14
•  What is periodic motion/oscillation
•  A(mplitude), (Period)T, Frequency(f), angular
frequency
•  Describing simple harmonic motion
•  Energy in simple harmonic motion
•  Pendulum: Simple pendulum and physical pendulum
•  (Optional: damped oscillations)
•  (Optional: driven oscillations and resonance)
Describing oscillations
Variables used to describe
oscillations:
•  Period: T
•  Frequency: f=1/T
•  Angular frenquency:
•  Amplitude: A
•  Maximum displacement
•  Where does maximum
speed occur?
Simple harmonic motion (SHM)
•  Simple F=-kx
When the restoring force is directly
proportional to the displacement from
equilibirum, the oscillation is called simple
Harmonic motion (SHM)
•  For a real spring,
Hookes’ Law is a good
approximation.
Deriving the period of a simple harmonic oscillation
A=0.20m, k=200N/m, m=0.5kg, T=?
Or Circular motion projection
Using integrals
dt =
dx
−v
1
1
1
 mv 2 = kA 2 − kx 2 ⇒ v =
2
2
2
m
k
∴dt = −
∫
T
4
0
dt =
0
∫A −
T π
⇒ =
4 2
k 2
(A − x 2 )
m
dx
A2 − x 2
m
k
v2
v2
Fx = −m cos θ = −m 2 x = −kx
A
A
k
→v =
A
m
dx
A2 − x 2
m
m
⇒ T = 2π
k
k
2πA
m
T=
= 2π
v
k
Restoring force!
Does not depend on A! (Similarly for simple pendulums)
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X versus t for SHM then simple variations on a theme
Circular motion, when projected on the x axis, is simply a SHM
x = Acosθ
θ = ωt
∴ x = Acos(ωt)
v
=
A
v x (t) = ?
ax (t) = ?
ω=
€
m
k
SHM phase, position, velocity, and acceleration
•  SHM can occur with
various phase angles
(initial displacement).
x = Acos(ωt)
•  X-t, v-t, and a-t with phase angles.
If starting from φ,
instead of 0
Once we have x(t), we
can determine v(t) and a(t)
x = Acos(ωt + φ )
How?
v = −ωAsin(ωt + φ )
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How?
a = −ω 2 Acos(ωt + φ )
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Energy in SHM
•  Energy is conserved during SHM
•  potential energy and kinetic energy interconvert
•  We know this; Just used it to derive the period!
•  Where do you find the maximum K, v, U, a,F, x
Q13.1
An object on the end of a spring is oscillating in simple harmonic
motion. If the amplitude of oscillation is doubled, how does this
affect the oscillation period T and the object’s maximum speed vmax?
A. T and vmax both double.
B. T remains the same and vmax doubles. C. T and vmax both remain the same. D. T doubles and vmax remains the same. E. T remains the same and vmax increases by a factor of
.
Q13.2
This is an x-t graph for
an object in simple
harmonic motion.
At which of the following times does the object have the most
negative velocity vx?
A. t = T/4
B. t = T/2
C. t = 3T/4
D. t = T
Q13.3
This is an x-t graph for
an object in simple
harmonic motion.
At which of the following times does the object have the most
negative acceleration ax?
A. t = T/4
B. t = T/2
C. t = 3T/4
D. t = T
Example
•  m is dropped on M.
•  What is the amplitude A,
and period T for
a)  Dropped at the
equilibrium position
b)  Dropped at the
maximum displacement
position
Q13.6
This is an x-t graph for
an object connected to
a spring and moving in
simple harmonic
motion.
At which of the following times is the potential energy
of the spring the greatest?
A. t = T/8
B. t = T/4
C. t = 3T/8
D. t = T/2 E. more than one of the above
Q13.7
This is an x-t graph for
an object connected to
a spring and moving in
simple harmonic
motion.
At which of the following times is the kinetic energy of
the object the greatest?
A. t = T/8
B. t = T/4
C. t = 3T/8
D. t = T/2 E. more than one of the above
Q13.8
To double the total energy of a mass-spring system
oscillating in simple harmonic motion, the amplitude
must increase by a factor of
A. 4.
B. C. 2. D. E. Applications of SHM: Vibrations of molecules
•  Atoms in crystals exert forces on their neighbors
•  How to describe their motions?
Vertical SHM
Vertical SHM has a different equilibrium position from
a horizontal SHM. Are their period Tv and TH any different?
Damped oscillations: Car shock absorbers
F = −kx − bv = ma
dx
d2x
−kx − b
=m 2
dt
dt
⇒ x = Ae −(b / 2m )t cos(ω ' t + φ )
with
ω'=
k
b2
−
m 4m 2
DampD(ecreaing)Amp(litude)
What happens if b is very large?
Overdamping:
x = C1e −a1 t + C2e −a 2 t
Critical damping: ω’=0
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Simple Pendulum
Small angle
Fθ = −mgsin θ ≈ −mgθ
approximation
x
Fθ = −mg
L
or
mg
Fx = −
x
L
m
L
⇒ T = 2π
= 2π
k = (mg /L)
g
Could you have guessed this?
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What is the initial angle is not small?
Grandpa’s pendulum clock and other planets
•  It’s actually expensive
•  For 300yrs, it was the most precise time
measurement instrument
•  Needs adjustment: g, L both changes. Why?
•  Would the clock be faster or slower on mars?
The physical pendulum
Comparisons:
F = −kx ⇒ T = 2π
m
k
τ = −kθ ⇒ T = 2π
I
k
 τ = −d(mg)sin θ ≈ −(dmg)θ
∴k = dmg
T = 2π
I
dmg
Example:
€
M=1kg
L=1m
T=?
This is how usually moment of inertia is measured.
Can you come up with the procedures?
Q13.9
A simple pendulum consists of a point mass
suspended by a massless, unstretchable string.
If the mass is doubled while the length of the string
remains the same, the period of the pendulum
A. becomes 4 times greater.
B. becomes twice as great. C. becomes greater by a factor of
D. remains unchanged.
E. decreases.
. €
Forced (driven) oscillations and resonance
•  A force applied “in synch” with a motion already in progress
will resonate and add energy to the oscillation
•  A singer can shatter a glass with a pure tone in tune with the
natural “ring” of a thin wine glass. (Will not be demonstrated
today
A=
Fmax
(k − mω d2 ) + b 2ω d2
Forced (driven) oscillations and resonance II
•  The Tacoma Narrows Bridge suffered spectacular structural
failure after absorbing too much resonant energy
Singer breaking glass
http://www.youtube.com/watch?v=amuPoPkAlx8
Summary
ω ,T, A,v, f
f = 1/T,ω = 2π /T
Basic variables:
SHM(Simple Harmonic Motions)
Energy conserved!
1
1
1
E = mv 2 + kx 2 = kA 2 = constan t
2
2
2
Oscillating springs:
ω=
€
k
m
T = 2π
Pendulum:ω =
g
L
T = 2€
π
L
g
Physical Pendulum:
ω=
mgd
I
T = 2π
€
€
m
k€
I
mgd
x = Acos(ωt + φ )
v = −ωAsin(ωt + φ )
a = −ω 2 Acos(ωt + φ )
CH15-17 (Not required) : A few useful concepts
Wave
•  How does Tsunami wave propagate?
Doppler Effect
•  How do you defend your traffic ticket?
Heat
Wave
Wave: A disturbance from equilibrium that propagates
Mechanical waves needs a medium
•  Earthquake, Sound, Ripple
Electromagnetic wave can travel in vacuum
Wave carries energy
v = λf = λ /T
Wave length
frequency
Understanding Tsunami
It’s a wave caused by a sudden displacement of huge
amount of water
Typical parameters:
It travels fast
What happens if it come
A: 1m
T: 20min
λ: 200km
ashore?v=?
Sound and Doppler effect
Sound is a longitudinal wave
•  Oscillation in the same direction as the propagation
Sound travels at the same speed in the same medium,
with different frequencies
Audible range: 20-20,000Hz
Doppler Effect
Doppler Effect
When a source and observer (listener) are in motion relative to each
other,the observed frequency is not the same as the source frequency!
v + vL
vL
fL =
= (1+ ) f S
λ
v
This effect occurs in any wave: Sound, radio wave (light, radar)
Police use radar to check your speed.
€
How to defend your speeding ticket?
Calibration?
Angle?
Factory parameters?
Interference?
Temperature and Heat
Celsius/Fahrenheit/Kelvin
5
TC = (TF − 32 o )
9
TK = TC + 273.15
1
3
m(v 2 ) av = kT
2
2
What is absolutely zero?
Laser cooling (trapping) of atoms can reach billionth above 0K
Heat: Energy transfer€due to temperature difference Q = mcΔT
Specific Heat: c
c water = 4190J /kg • K
c Iron = 470J /kg • K
Phase change: transition between different€states of matter
(solid, gas, liquid)
Q = mL f€/ v
Latent Heat of fusion/vaporization
€
€
L f / water = 334000J /kg
Lv / water = 2256000J /kg
Preparing for the final
Start from the summary page at end of chapters!
Concentrate on problems from the Exams, problems
explained in lectures, and problems from
homework!
Use my office hour, or come at any time by
appointment!
Good luck everyone!