Download (x = 0 ).

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m
AP Physics Lecture Notes
Vibrations and Waves
Units of Chapter 11
Simple Harmonic Motion
Energy in the Simple Harmonic Oscillator
The Period and Sinusoidal Nature of SHM
The Simple Pendulum
Vibrations and Waves
Simple Harmonic Motion
If an object vibrates or oscillates back and forth over the
same path, each cycle taking the same amount of time, the
motion is called periodic (T).
m
X=0
We assume that the surface is frictionless. There is a point
where the spring is neither stretched nor compressed; this
is the equilibrium position. We measure displacement
from that point (x = 0 ).
Vibrations and Waves
Simple Harmonic Motion
m
x=0
F
m
x
The force exerted by the spring depends on the displacement:
F  kx
The minus sign on the force indicates that it is a restoring
force – it is directed to restore the mass to its equilibrium
position.
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Simple Harmonic Motion
F
m
F  kx
x
(a) (k) is the spring constant
(b) Displacement (x) is measured from the equilibrium point
(c) Amplitude (A) is the maximum displacement
(d) A cycle is a full to-and-fro motion
(e) Period (T) is the time required to complete one cycle
(f) Frequency (f) is the number of cycles completed per second
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Simple Harmonic Motion
If the spring is hung vertically, the only
change is in the equilibrium position,
which is at the point where the spring
force equals the gravitational force.
F  kxo
xo
m
Equilibrium
Position
mg
Vibrations and Waves
11-1 Simple Harmonic Motion
Any vibrating system where the restoring force is
proportional to the negative of the displacement
F  kx
moves with simple harmonic motion (SHM), and is often
called a simple harmonic oscillator.
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Energy in the Simple Harmonic Oscillator
Potential energy of a spring is given by:
kx2
PE 
2
The total mechanical energy is then:
mv 2 kx2
Etotal  

2
2
The total mechanical energy will be conserved
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Energy in the Simple Harmonic Oscillator
m
A
If the mass is at the limits of its motion, the
energy is all potential.
kA 2
PE 
2
vma
x
m
2
mv max
KE 
2
x=0
If the mass is at the equilibrium point, the
energy is all kinetic.
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Energy in the Simple Harmonic Oscillator
kA 2
Etotal  
2
The total energy is, therefore
And we can write:
kA 2 mv 2 kx2


2
2
2
This can be solved for the velocity as a function of position:

k 2
v
A  x2
m
where

 
k 2
k
v max 
A A
m
m
Vibrations and Waves
The Period and Sinusoidal Nature of SHM
vma
If we look at the projection onto the
x axis of an object moving in a circle
of radius A at a constant speed vmax,
we find that the x component of its
velocity varies as:
v  v max sin θ

k  A 2  x 2 
v  A


m 
A



k 2
v
A  x2
m

This is identical to SHM.
v x
A
q
A2  x2
x
k
v max  A
m
A2  x2
sin θ 
A
Vibrations and Waves
The Period and Sinusoidal Nature of SHM
Therefore, we can use the period and frequency of a particle
moving in a circle to find the period and frequency:
2 πA
k

v max  A
T
m
k
v max  A
m
m
T  2π
k
1 1 k
f 
T 2π m
Vibrations and Waves
The Period and Sinusoidal Nature of SHM
The acceleration can be calculated as function of displacement
F
m
x
F  kx
ma  kx
k
a    x
m
k
amax     A
m
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The Simple Pendulum
A simple pendulum consists of a mass at the end of a
lightweight cord. We assume that the cord does not
stretch, and that its mass is negligible.
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Simple Pendulum
F  mg sinq 
q
L
 mg 
F  
x
 L 
Small angles x  s
 mg 
F  
s
 L 
m
T  2
k
x
sinq  
L
m
 2
mg
L
k for
SHM
x
m
F
s
q
mg
L
 T  2
g
Vibrations and Waves