Chapter 13: Vibrations and Waves

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Chapter 13 Vibrations and Waves
Suggested homework problems12,33,47,54,58

• Hookes Law

• Hookes law and oscillation

• A simple example of vibration motion an object
  attached to a spring.

No friction
The negative sign means that the force exerted by
the spring is always directed opposite the
displacement of the object (restoring force).
A restoring force always pushes or pulls the
object toward the equilibrium position.
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• Hookes Law

• Hookes law and oscillation (contd)

No friction

• A simple example of vibration motion
• an object attached to a spring.

Suppose the object is initially pulled a distance
A to the right and released from rest. Then the
object does simple harmonic motion. Simple
harmonic motion occurs when the net force along
the direction of motion obeys Hookes law when
the net force is proportional to the displacement
from equilibrium point and is always directed
toward the equilibrium point.
periodic motion
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• Hookes Law

• Hookes law and oscillation (contd)

No friction

• A simple example of vibration motion
• an object attached to a spring.

Terminology - The amplitude A is the maximum
distance of the object from its equi-
librium position. Alt x lt A. - The period T
is the time it takes the object to move
through one complete cycle of motion from xA
to x-A and then back to xA. - The
frequency f is the number of complete cycles
or vibrations per unit time. f1/T.
periodic motion
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• Hookes Law

• Hookes law and oscillation (contd)

• Example 13.1 Measuring the spring constant

• Harmonic oscillator equation

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• Elastic Potential Energy

• Elastic potential energy

• The energy stored in a stretched or compressed
  spring

• The conservation of energy

If there are only conservative forces
If there are also non-conservative forces
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• Elastic Potential Energy

• Elastic potential energy

• Conservation of energy

- The velocity is zero at xA,-A. - The velocity
is at its maximum at x0
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• Comparing Simple Harmonic Motion with
• Uniform Circular Motion

• Uniform circular motion

• A circular motion and its projection

As the turntable rotates with constant angular
speed, the shadow of the ball moves back and
forth with simple harmonic motion.
c.f. simple harmonic motion
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• Comparing Simple Harmonic Motion with
• Uniform Circular Motion

• Period and Frequency

• Period of oscillation (T)

One period is completed when the ball rotates
360o and moves a distance v0T.
From conservation of energy, for simple harmonic
oscillation of a spring system with the spring
constant k at x0,
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• Comparing Simple Harmonic Motion with
• Uniform Circular Motion

• Period and Frequency

• Frequency (f) and angular frequency (w)

Frequency is how many complete rotations/cycles a
simple harmonic oscillation or uniform circular
motion makes per unit time.
Units hertz (Hz) cycles per second
Angular frequency is a frequency measured in
terms of angle.
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• Position, Velocity, and Acceleration
• as a Function of Time

• x vs. time

• We can obtain an expression for the position of
  an object with
• simple harmonic motion as a function of time.

if constant angular speed
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• Position, Velocity, and Acceleration
• as a Function of Time

• v vs. t

• We can obtain an expression for the velocity of
  an object with
• simple harmonic motion as a function of time.

if constant angular speed
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• Position, Velocity, and Acceleration
• as a Function of Time

• Period and Frequency

• We can obtain an expression for the position of
  an object with
• simple harmonic motion as a function of time.

sinusoidal
if constant angular speed
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• Motion of a Pendulum

• Pendulum

• If a force is a restoring one, from an analogy
  of a Hookes law
• we can prove that the system under influence
  the force makes
• simple harmonic oscillation.

In an analogy to Hookes law Ft-kx,
Ft
The motion of a pendulum is not simple harmonic
in general but it is if the angle q is small.
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• Motion of a Pendulum

• Physical pendulum

• In general case, the argument for a pendulum
  system of a mass
• attached to a string can be used to an object
  of any shape.

I moment of inertia
For a simple pendulum,
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• Damped Oscillation

• Oscillation with friction

• In any real systems, forces of frictions retard
  the motion induced
• by restoring forces and the system do not
  oscillate indefinitely.

The friction reduces the mechanical energy of the
system as time passes, and the motion is said to
be damped.
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• Waves

• Examples and sources of waves

• The world is full of waves sound waves, waves
  on a string,
• seismic waves, and electromagnetic waves such
  as light, radio
• waves, TV signals, x-rays, and g-rays.

• Waves are produced by some sort of vibration

Vibration of vocal cords, guitar strings, etc
sound
Vibration of electrons in an antenna, etc
radio waves
water waves
Vibration of water

• Types of waves

• Transverse waves

• The bump (pulse) travels to the right
• with a definite speed traveling wave

• Each segment of the rope that is disturbed
• moves in a direction perpendicular to the
• wave motion transverse wave

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• Waves

• Types of waves (contd)

• Longitudinal waves

• The elements of the medium undergo displacements
  parallel
• to the direction of wave motion longitudial
  wave

• Their disturbance corresponds to a series of
  high- and low-
• pressure regions that may travel through air
  or through any
• material medium with a certain speed.

sound wave longitudinal wave
C compression R rarefaction
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• Waves

• Types of waves (contd)

• Longitudinal-transverse waves

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• Frequency, Amplitude, and Wavelength

• Frequency, amplitude, and wavelength

• Consider a string with one end connected to a
  blade vibrating
• according to simple harmonic oscillation.

Amplitude A The maximum distance the string
moves.
Wavelength l The distance between two
successive crests
Wave speed v vDx/Dtl/T (wavelength/period)
Frequency f vl/Tfl (wavelength/period)
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• Frequency, Amplitude, and Wavelength

• Examples

• Example 13.8 A traveling wave

A wave traveling in the positive x-direction.
Find the amplitude, wavelength, speed, and period
of the wave if it has a frequency of 8.00 Hz.
Dx40.0 cm and Dy15.0 cm.
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Frequency, Amplitude, and Wavelength

• Examples (contd)

• Example 13.9 Sound and light

A wave has a wavelength of 3.00 m. Calculate the
frequency of the wave if it is (a) a sound wave,
and (b) a light wave. Take the speed of sound as
343 m/s and that of light as 3.00x108 m/s. (a)
(b)
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• Speed of Waves on Strings

• Speed of waves on strings

• Two types of speed

• The speed of the physical string that vibrates
  up and down
• transverse to the string in the y-direction

• The rate at which the disturbance propagates
  along the length
• of the string in the x-direction wave speed

• For a fixed wavelength, a string under greater
  tension F has a
• greater wave speed because the period of
  vibration is shorter,
• and the wave advances one wavelength during one
  period.
• A string with greater mass per unit length m
  (linear density) vibrates
• more slowly, leading to a longer period and a
  slower wave speed.

Dimension analysis FML/T2, mM/L,
F/mL2/T2, qF/mL/Tv
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• Speed of Waves on Strings

• Example 13.10

• A uniform string has a mass M of 0.0300 kg and a
  length L of 6.00 m.
• Tension is maintained in the string by
  suspending a block of mass
• m 2.00 kg from one end.

(a) Find the speed of the wave.
(b) Find the time it takes the pulse to
travel from the wall to the pulley.
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• Interference of Waves

• Superposition principle

• Tow traveling waves can meet and pass through
  each other without
• being destroyed or even altered.

• When two or more raveling waves encounter each
  other while
• moving through a medium, the resultant wave is
  found by adding
• together the displacements of the individual
  waves point by point.

• Interference

constructive interference (in phase)
destructive interference (out of phase)
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• Interference of Waves

• Example 13.10

• A uniform string has a mass M of 0.0300 kg and a
  length L of 6.00 m.
• Tension is maintained in the string by
  suspending a block of mass
• m 2.00 kg from one end.

(a) Find the speed of the wave.
(b) Find the time it takes the pulse to
travel from the wall to the pulley.
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• Reflection of Waves

• Reflection of waves at a fixed end

Reflected wave is inverted
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• Reflection of Waves

• Reflection of waves at a free end

Reflected wave is not inverted