### 01 Simple Harmonic Motion

```Ch 17: Simple Harmonic
Motion
M Sittig
AP Physics B
Summer Course 2012
2012年AP物理B暑假班
Simple Harmonic Motion
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Humans are machines for identifying
patterns, it’s fundamental to how our brains
work.
SHM is a repeating pattern in motion,
describes many patterns we see in nature.
Covered at a basic level on the AP test, but
has lots of vocabulary.
Oscillation
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Medium
Disturbance
Restoring force
Examples:
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Ocean waves
Pendulum
Spring
Cycle, Amplitude, Period
Period, Frequency
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Period (T): The time to complete one cycle.
Frequency (f): The number of cycles in one
second.
1
f 
T
1
T
f
Period and frequency
Period (s)
1
T
f
Frequency
(1/s or Hz)
Practice Problem
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A girl in boat watches waves on a lake
passing with a half-second pause between
each crest. She notices that each wave takes
1.5 s to sweep straight down the length of
her 4.5 m boat.
Determine
a) the period of the waves
b) the frequency of the waves
ConcepTest 11.1a Harmonic Motion I
A mass on a spring in SHM has
1) 0
amplitude A and period T. What
2) A/2
is the total distance traveled by
3) A
the mass after a time interval T?
4) 2A
5) 4A
ConcepTest 11.1a Harmonic Motion I
A mass on a spring in SHM has
1) 0
amplitude A and period T. What
2) A/2
is the total distance traveled by
3) A
the mass after a time interval T?
4) 2A
5) 4A
In the time interval T (the period), the mass goes
through one complete oscillation back to the starting
point. The distance it covers is: A + A + A + A (4A).
ConcepTest 11.1b Harmonic Motion II
A mass on a spring in SHM has
amplitude A and period T. What is
the net displacement of the mass
after a time interval T?
1) 0
2) A/2
3) A
4) 2A
5) 4A
ConcepTest 11.1b Harmonic Motion II
A mass on a spring in SHM has
amplitude A and period T. What is
the net displacement of the mass
after a time interval T?
1) 0
2) A/2
3) A
4) 2A
5) 4A
The displacement is Dx = x2–x1. Since the
initial and final positions of the mass are the
same (it ends up back at its original position),
then the displacement is zero.
Follow-up: What is the net displacement after a half of a period?
ConcepTest 11.1c Harmonic Motion III
A mass on a spring in SHM has
amplitude A and period T. How
long does it take for the mass to
travel a total distance of 6A?
1) 1/2 T
2) 3/4 T
3) 1 1/4 T
4) 1 1/2 T
5) 2 T
ConcepTest 11.1c Harmonic Motion III
A mass on a spring in SHM has
amplitude A and period T. How
long does it take for the mass to
travel a total distance of 6A?
1) 1/2 T
2) 3/4 T
3) 1 1/4 T
4) 1 1/2 T
5) 2 T
We have already seen that it takes one period T to travel a total
distance of 4A. An additional 2A requires half a period, so the total
time needed for a total distance of 6A is 1 1/2 T.
Follow-up: What is the net displacement at this particular time?
Vibrating Mass on a Spring
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The radius of a circle is symbolic of
the amplitude of a wave.
Energy is conserved as the elastic
potential energy in a spring can be
converted into kinetic energy. Once
again the displacement of a spring
is symbolic of the amplitude of a
wave
Since BOTH algebraic expressions
have the ratio of the Amplitude to
the velocity we can set them equal
to each other.
This derives the PERIOD of a
SPRING.
Period of a mass on a spring
Period (s)
m
T  2
k
Spring
constant
(N/m)
Mass (kg)
Example Problem
Practice Problem
Simple Pendulum
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Derivation is a bit complicated, but I’ll
include it here:
Pendulums
Consider the FBD for a pendulum. Here we have the
weight and tension. Even though the weight isn’t at an
angle let’s draw an axis along the tension.
q
q
mgcosq
mgsinq
m gsin q  RestoringForce
m gsin q  kx
Pendulums
s s
q 
R L
s  qL  Am plitude
m gsin q  RestoringForce
m gsin q  kx
m g sin q  kqL
sin q  q , if q  sm all
m g  kl
m l

k g
Tspring
m
 2
k
What is x? It is the
amplitude! In the picture to
the left, it represents the
chord from where it was
released to the bottom of
the swing (equilibrium
position).
Tpendulum
l
 2
g
Period of a pendulum
Period (s)
l
T  2
g
Acceleration
due to gravity
(m/s2)
Length of
pendulum
(m)
Example Problem
Practice Problem
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A visitor to a lighthouse wishes
to determine the height of the
tower. She ties a spool of thread
to a small rock to make a simple
pendulum, which she hangs
down the center of a spiral
staircase of the tower. The
period of oscillation is 9.40 s.
What is the height of the tower?
ConcepTest 11.5a Energy in SHM I
A mass oscillates in simple
harmonic motion with amplitude
A. If the mass is doubled, but the
amplitude is not changed, what
will happen to the total energy of
the system?
1) total energy will increase
2) total energy will not change
3) total energy will decrease
ConcepTest 11.5a Energy in SHM I
A mass oscillates in simple
harmonic motion with amplitude
A. If the mass is doubled, but the
amplitude is not changed, what
will happen to the total energy of
the system?
1) total energy will increase
2) total energy will not change
3) total energy will decrease
The total energy is equal to the initial value of the
elastic potential energy, which is PEs = 1/2 kA2. This
does not depend on mass, so a change in mass will
not affect the energy of the system.
Follow-up: What happens if you double the amplitude?
ConcepTest 11.6a Period of a Spring I
A glider with a spring attached to
each end oscillates with a certain
period. If the mass of the glider is
doubled, what will happen to the
period?
1) period will increase
2) period will not change
3) period will decrease
ConcepTest 11.6a Period of a Spring I
A glider with a spring attached to
each end oscillates with a certain
period. If the mass of the glider is
doubled, what will happen to the
period?
1) period will increase
2) period will not change
3) period will decrease
The period is proportional to the
square root of the mass. So an
increase in mass will lead to an
increase in the period of motion.
T = 2
(m/k)
Follow-up: What happens if the amplitude is doubled?
ConcepTest 11.7a Spring in an Elevator I
A mass is suspended from the
ceiling of an elevator by a spring.
When the elevator is at rest, the
period is T. What happens to the
period when the elevator is moving
upward at constant speed?
1) period will increase
2) period will not change
3) period will decrease
ConcepTest 11.7a Spring in an Elevator I
A mass is suspended from the
ceiling of an elevator by a spring.
When the elevator is at rest, the
period is T. What happens to the
period when the elevator is moving
upward at constant speed?
1) period will increase
2) period will not change
3) period will decrease
Nothing at all changes when the elevator moves at constant
speed. The equilibrium elongation of the spring is the same,
and the period of simple harmonic motion is the same.
ConcepTest 11.7b Spring in an Elevator II
A mass is suspended from the
ceiling of an elevator by a spring.
When the elevator is at rest, the
period is T. What happens to the
period when the elevator is
accelerating upward?
1) period will increase
2) period will not change
3) period will decrease
ConcepTest 11.7b Spring in an Elevator II
A mass is suspended from the
ceiling of an elevator by a spring.
When the elevator is at rest, the
period is T. What happens to the
period when the elevator is
accelerating upward?
1) period will increase
2) period will not change
3) period will decrease
When the elevator accelerates upward, the hanging mass feels
“heavier” and the spring will stretch a bit more. Thus, the
equilibrium elongation of the spring will increase. However, the
period of simple harmonic motion does not depend upon the
elongation of the spring – it only depends on the mass and the
spring constant, and neither one of them has changed.
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