Simple harmonic motion PowerPoint presentation

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Lecture PowerPoints
Chapter 11
Physics: Principles with
Applications, 7th edition
Giancoli

Chapter 11
Oscillations and Waves

Contents of Chapter 11
• Simple Harmonic Motion—Spring Oscillations
• Energy in Simple Harmonic Motion
• The Period and Sinusoidal Nature of SHM
• The Simple Pendulum
• Damped Harmonic Motion
• Forced Oscillations; Resonance
• Wave Motion
• Types of Waves and Their Speeds: Transverse and
Longitudinal

Contents of Chapter 11
• Energy Transported by Waves
• Reflection and Transmission of Waves
• Interference; Principle of Superposition
• Standing Waves; Resonance
• Refraction
• Diffraction
• Mathematical Representation of a Traveling Wave

11-1 Simple Harmonic Motion—Spring
Oscillations
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. The mass
and spring system is a
useful model for a periodic
system.

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 on the
previous figure).
The force exerted by the spring depends on the
displacement:
Oscillations
(11-1)

Oscillations
• The minus sign on the force indicates that it is a
restoring force—it is directed to restore the mass to
its equilibrium position.
• k is the spring constant
• The force is not constant, so the acceleration is not
constant either

Oscillations
• Displacement is measured from the
equilibrium point
• Amplitude is the maximum
displacement
• A cycle is a full to-and-fro motion;
this figure shows half a cycle
• Period is the time required to
complete one cycle
• Frequency is the number of cycles
completed per second

Oscillations
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.

Oscillations
Any vibrating system where the restoring force is
proportional to the negative of the displacement is in
simple harmonic motion (SHM), and is often called a
simple harmonic oscillator.

11-2 Energy in Simple Harmonic Motion
We already know that the potential energy of a spring is
given by:
PE = ½ kx2
The total mechanical energy is then:
The total mechanical energy will be conserved, as we are
assuming the system is frictionless.
(11-3)

If the mass is at the limits of its
motion, the energy is all potential.
If the mass is at the equilibrium
point, the energy is all kinetic.
We know what the potential energy
is at the turning points:
(11-4a)

The total energy is, therefore ½ kA2
And we can write:
This can be solved for the velocity as a function of
position:
where
(11-4c)
(11-5b)
(11-5a)

11-3 The Period and Sinusoidal Nature
of SHM
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:
This is identical to SHM.
(11-5b)

of SHM
Therefore, we can use the period and frequency of a
particle moving in a circle to find the period and
frequency:
(11-6b)
(11-6a)

of SHM
We can similarly find the position as a function of time:
(11-8c)
(11-8b)
(11-8a)

of SHM
The top curve is a graph of
the previous equation.
The bottom curve is the
same, but shifted ¼ period
so that it is a sine function
rather than a cosine.

of SHM
The velocity and acceleration can
be calculated as functions of
time; the results are below, and
are plotted at left.
(11-10)
(11-9)

11-4 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.

In order to be in SHM, the restoring force
must be proportional to the negative of the
displacement. Here we have F = -mg sin θ
which is proportional to sin θ and not to
θ itself.
However, if the angle
is small, sin θ ≈ θ.

Therefore, for small angles, the force is approximately
proportional to the angular displacement.
The period and frequency are:
(11-11a)
(11-11b)

So, as long as the cord can be
considered massless and the
amplitude is small, the period
does not depend on the mass.

11-5 Damped Harmonic Motion
Damped harmonic motion is harmonic motion with a
frictional or drag force. If the damping is small, we can
treat it as an “envelope” that modifies the undamped
oscillation.

However, if the damping is
large, it no longer
resembles SHM at all.
A: underdamping: there
are a few small oscillations
before the oscillator comes
to rest.
B: critical damping: this is the fastest way to get to
equilibrium.
C: overdamping: the system is slowed so much that it takes a
long time to get to equilibrium.

There are systems where damping is unwanted, such as clocks and
watches.
Then there are systems in which it is
wanted, and often needs to be as close to
critical damping
as possible, such
as automobile
shock absorbers
and earthquake
protection for
buildings.

Forced vibrations occur when there is a periodic driving
force. This force may or may not have the same period
as the natural frequency of the system.
If the frequency is the same as the natural frequency, the
amplitude becomes quite large. This is called resonance.
11-6 Forced Oscillations; Resonance

The sharpness of the
resonant peak depends on
the damping. If the
damping is small (A), it
can be quite sharp; if the
damping is larger (B), it is
less sharp.
Like damping, resonance can be wanted or unwanted.
Musical instruments and TV/radio receivers depend
on it.
11-6 Forced Oscillations; Resonance

11-7 Wave Motion
A wave travels
along its medium,
but the individual
particles just move
up and down.

All types of traveling waves transport energy.
Study of a single wave
pulse shows that it is begun
with a vibration and
transmitted through internal
forces in the medium.
Continuous waves start
with vibrations too. If the
vibration is SHM, then the
wave will be sinusoidal.
11-7 Wave Motion

11-7 Wave Motion
Wave characteristics:
• Amplitude, A
• Wavelength, λ
• Frequency f and period T
• Wave velocity
(11-12)

11-8 Types of Waves and Their Speeds:
Transverse and Longitudinal
The motion of particles in a wave can either be
perpendicular to the wave direction (transverse) or
parallel to it (longitudinal).

11-8 Types of Waves and Their Speeds:
Sound waves are longitudinal waves:

11-8 Types of Waves and Their Speeds :
Earthquakes produce both longitudinal and transverse waves.
Both types can travel through solid material, but only
longitudinal waves can propagate through a fluid—in the
transverse direction, a fluid has no restoring force.
Surface waves are waves that travel along the boundary
between two media.

11-9 Energy Transported by Waves
Just as with the oscillation that starts it, the energy
transported by a wave is proportional to the square of the
amplitude.
Definition of intensity:
The intensity is also proportional to the square of the
amplitude:
(11-15)

If a wave is able to spread out three-dimensionally from
its source, and the medium is uniform, the wave is
spherical.
Just from geometrical
considerations, as long as
the power output is constant,
we see:
(11-16b)

By looking at the energy
of a particle of matter in
the medium of the wave,
we find:
Then, assuming the entire medium has the same density, we
find:
Therefore, the intensity is proportional to the square of the
frequency and to the square of the amplitude.
(11-17a)
(11-18)

11-10 Reflection and Transmission of Waves
A wave reaching the
end of its medium, but
where the medium is
still free to move, will
be reflected (b), and its
reflection will be
upright.
A wave hitting an obstacle will be reflected (a), and its
reflection will be inverted.

A wave encountering
a denser medium will
be partly reflected and
partly transmitted; if
the wave speed is less
in the denser medium,
the wavelength will
be shorter.

Two- or three-dimensional waves can be represented by
wave fronts, which are curves of surfaces where all the
waves have the same phase.
Lines perpendicular to the
wave fronts are called rays;
they point in the direction
of propagation of the wave.

The law of reflection: the angle of incidence equals the
angle of reflection.

11-11 Interference; Principle of Superposition
The superposition principle says that when two waves pass
through the same point, the displacement is the arithmetic sum
of the individual displacements.
In the figure below, (a) exhibits destructive interference and (b)
exhibits constructive interference.

11-11 Interference; Principle of Superposition
These figures show the sum of two waves. In (a) they
add constructively; in (b) they add destructively; and in
(c) they add partially destructively.

11-12 Standing Waves; Resonance
Standing waves occur when
both ends of a string are
fixed. In that case, only
waves which are motionless
at the ends of the string can
persist. There are nodes,
where the amplitude is
always zero, and antinodes,
where the amplitude varies
from zero to the maximum
value.

The frequencies of the
standing waves on a
particular string are called
resonant frequencies.
They are also referred to as
the fundamental and harmonics.

The wavelengths and frequencies of standing waves are:
(11-19a)
(11-19b)

11-13 Refraction
If the wave enters a medium where the wave speed is
different, it will be refracted—its wave fronts and rays
will change direction.
We can calculate the angle
of refraction, which depends
on both wave speeds:
(11-20)

11-13 Refraction
The law of refraction works
both ways—a wave going
from a slower medium to
a faster one would follow
the red line in the
other direction.

11-14 Diffraction
When waves encounter an
obstacle, they bend around
it, leaving a “shadow
region.” This is called
diffraction.

11-14 Diffraction
The amount of diffraction depends on the size of the
obstacle compared to the wavelength. If the obstacle is
much smaller than the wavelength, the wave is barely
affected (a). If the object is comparable to, or larger than,
the wavelength, diffraction is much more significant
(b, c, d).

11-15 Mathematical Representation of a
Traveling Wave
To the left, we have a
snapshot of a traveling
wave at a single point
in time. Below left, the
same wave is shown
traveling.

11-15 Mathematical Representation of a
Traveling Wave
A full mathematical description of the wave describes
the displacement of any point as a function of both
distance and time:
(11-22)

Summary of Chapter 11
• For SHM, the restoring force is proportional to the
displacement.
• The period is the time required for one cycle, and the
frequency is the number of cycles per second.
• Period for a mass on a spring:
• SHM is sinusoidal.
• During SHM, the total energy is continually changing
from kinetic to potential and back.
(11-6a)

• A simple pendulum approximates SHM if its amplitude
is not large. Its period in that case is:
• When friction is present, the motion is damped.
• If an oscillating force is applied to a SHO, its amplitude
depends on how close to the natural frequency the
driving frequency is. If it is close, the amplitude
becomes quite large. This is called resonance.
(11-11a)

• Vibrating objects are sources of waves, which may be
either a pulse or continuous.
• Wavelength: distance between successive crests.
• Frequency: number of crests that pass a given point
per unit time.
• Amplitude: maximum height of crest.
• Wave velocity: v = λf

• Transverse wave: oscillations perpendicular to direction
of wave motion.
• Longitudinal wave: oscillations parallel to direction of
wave motion.
• Intensity: energy per unit time crossing unit area (W/m2):
• Angle of reflection is equal to angle of incidence.
(11-16b)

• When two waves pass through the same region of space,
they interfere. Interference may be either constructive or
destructive.
• Standing waves can be produced on a string with both
ends fixed. The waves that persist are at the resonant
frequencies.
• Nodes occur where there is no motion; antinodes where
the amplitude is maximum.
• Waves refract when entering a medium of different wave
speed, and diffract around obstacles.

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