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Last Lecture Quiz Thursday on simple pendula. You should review Example 10.9
Quiz and homework grades will be uploaded to Sakai.
Test 3 breakdown next time. Exam 3 is Friday, August 3.
We will meet for 6 lecture meetings. Included are 2 exam reviews and the final exam.
Start Chapter 11 today.
Last time we learned that a rigid body does not exist in nature. All object deform when
stressed. In the elastic limit the more general form of Hooke’s law holds and
stress ∝ strain
Simple Harmonic Motion
Vibration is repeated motion back and forth along the same path. Vibrations occur in the
vicinity of a point of stable equilibrium.
An equilibrium points is stable is the net force on the object when it is displaced a
small distance from equilibrium points back towards the equilibrium point.
This type of force is called a restoring force. Simple harmonic motion (SHM)
occurs whenever the restoring force if proportional to the displacement from equilibrium.
Simple harmonic motion can be used to approximate small vibrations.
Energy Analysis
For a spring-mass system, we know that energy is conserved.
E = K + U = constant
Recalling the definitions for K and U
E = 12 mv 2 + 12 kx 2
We will call the maximum displacement of the body the amplitude A. At the
maximum displacement, the object stops.
Etotal = 12 m(0)2 + 12 kA2 = 12 kA2
The maximum speed occurs at equilibrium (x = 0) and U = 0
Etotal = 12 mvm + 12 k (0) 2 = 12 mvm
2
2
Equating the two forms of Etotal gives
vm =
k
A
m
Acceleration in SHM
From Hooke’s law for springs
Fx = −kx
max = −kx
ax =
− kx
m
The acceleration is proportional to the displacement and in the opposite direction. The
largest acceleration occurs at the largest displacement (A)
ax =
kA
m
Period and Frequency
One cycle means that the particle is at the same location and heading the same direction.
The period (T) is the time to complete one cycle. The frequency (f) is the number of
cycles per second. Just as before
f =
1
T
Not all vibrations are simple harmonic.
Circular Motion and Simple Harmonic Motion
The projection of uniform circular motion along any axis (the x-axis here) is the same as
simple harmonic motion. We use our understanding of uniform circular motion to arrive
at the equations of simple harmonic motion.
The projection of the position on the x-axis gives
x(t ) = A cos ωt
The acceleration is inward and
a (t ) = −ω 2 A cos ωt
The velocity can be shown to be
v(t ) = −ωA sin ωt
The angular frequency is
ω=
k
m
For the ideal spring mass system
f =
1
ω
=
2π 2π
k
m
T=
1
m
= 2π
f
k
How do you keep these straight? Memorize one and derive the other.
The Pendulum
For small oscillations, the simple pendulum executes simple harmonic motion.
A simple pendulum is point mass suspended from a string.
Applying Newton’s second law
∑F
x
= ma x
− T sin θ = ma x
−T
x
= ma x
L
For small angles sin θ = θ. The acceleration along the y-axis is negligible
∑F
y
= ma y
T cos θ − mg = 0
T = mg
For small angles cos θ = 1. Substituting for T
x
= max
L
x
− mg = max
L
g
ax = − x
L
−T
This is the condition of simple harmonic motion. We can pick off the angular frequency
as
ω=
g
L
The period of the pendulum
T = 2π
L
g
Physical Pendulum.
Instead of a point mass, the vibrating object has size.
The period is
T = 2π
I
mgd
where d is the distance from the rotation axis to the center of mass of the object and I is
the rotational inertia about the rotation axis (not the center of mass).
Damped Oscillations
In reality a vibrating system will stop vibrating. Energy is lost to the surroundings. Since
it is similar to the drag felts when moving through water, it is called damped motion.
Forced Oscillations and Resonance
To compensate for energy lost in damped systems, it is possible to drive the system at its
resonant frequency.
Forced oscillations can occur when the periodic driving force acts on a system
that can vibrate. If the system is driven at its resonance frequency, vibrations can
increase in amplitude until the system is destroyed.
Chapter 11 Waves
Energy can be transported by particles or waves
A wave is characterized as some sort of disturbance that travels away from a source. The
key difference between particles and waves is a wave can transmit energy from one point
to another without transporting any matter between the two points.
The intensity is the average power per unit area. It is measured in W/m2.
I=
P
A
As you move away from the source, the intensity drops off
I=
P
P
=
A 4πr 2
Two types of waves
• Transverse – the motion of the particle in the medium are perpendicular to the
direction of propagation of the wave. (wave in string, electromagnetic)
• Longitudinal – the motion of the particles in the medium are along the same line
as the direction of the wave. (sound)