Holt Physics Chapter 11

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Holt Physics Chapter 11

• Vibrations and Waves

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Simple Harmonic Motion

• Simple Harmonic Motion vibration about an
  equilibrium position in which a restoring force
  is proportional to the displacement from
  equilibrium.
• Springs, pendulums, etc
• The spring force always pushes or pulls the mass
  back toward its original equilibrium position
  (sometimes called a restoring force).
• See figure 12-1, page 438

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Vibrations and Waves
Simple Harmonic Motion
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Example gravity acting on a mass hanging from a
string.

Example gravity acting on a mass hanging from a
spring.
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When the restoring force is linearly proportional
to the amount of the displacement from
equilibrium, the force is said to be a Hookes
Law force.
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Hookes Law

• For mass-spring systems
• Formula Felastic -kx
• Spring force-(spring constantXdisplacement)
• The negative sign in the equation refers to the
  force being opposite to the displacement.
• k is a measure of stiffness and the unit is N/m
• Remember, stretching or compressing a spring
  stores elastic potential energy that can be
  converted to kinetic energy.

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Book Example, page 366

• If a mass of 0.55 kg attached to a vertical
  spring stretches the spring 2.0 cm from its
  original equilibrium position, what is the spring
  constant?
• m0.55 kg x-2.0cm -0.020m
• g9.81m/s2 k?

Elastic force is up () and displacement (x)
is down (-)
They are always opposite...
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If a mass of 0.55 kg attached to a vertical
spring stretches the spring 2.0 cm from its
original equilibrium position, what is the spring
constant?

• m0.55 kg x-2.0cm -0.020m
• g9.81m/s2 k? Felastic mg
• Felastic -kx
• mg-kx
• k-mg/x
• k-(0.55kg)(9.81m/s2)/(-.020m)
• k270N/m

Felastic
No negative!!
Fg
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Example 2. How much force will a vertical
spring with a spring constant of 350 N/m exert
when it is stretched to 22 cm past its
equilibrium point?
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Example 3. When a 1.5 kg mass is suspended from
a spring hanging vertically, How much does it
stretch if the spring constant Is 98N/m?
Remember The spring force (FE) is equal and
opposite to the weight (Fw).
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• Practice page 367

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Simple Harmonic Motion, pg. 371
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• Some basic elements of simple harmonic motion
• At equilibrium position, velocity maximum
• At maximum displacement, the spring force and
  acceleration maximum
• This motion repeats itself in simple harmonic
  motion.

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• See page 371 for a comparison of spring-mass
  systems and pendulum motion.
• The pendulum stores gravitational potential
  energy, and the spring-mass system stores elastic
  potential energy.
• See page 370

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Measuring Simple Harmonic Motion

• Amplitude maximum displacement from equilibrium
  position.
• Unit is radians or meters
• Period (T) time it takes to complete one cycle
  of motion.
• Unit is seconds
• Frequency (f) is the number of cycles per second.
  T1/f or f1/T
• Unit for f is s-1 or Hertz (Hz)

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Calculating the Period of a Simple Pendulum

• T 2? L/g
• L length of pendulum
• g free-fall
• T Period of pendulum
• Amplitude and mass do not affect the period of a
  pendulum

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Measuring Simple Harmonic Motion
Example 1. A chandelier exhibiting
simple harmonic motion has a period of 1.3 s.
How long is the cord it hangs from?
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Example 2. An astronaut on a foreign planet
notes that a pendulum with a length of 2.5 m has
a period of 1.4 s. Find g on this planet.
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Example 3. Find the period (T) and the
frequency (f) for wave (B) below.
From the image, we see that the period, or time
to complete one cycle, is 0.60 s.
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Example 4. Find the period (T) and the
frequency (f) for wave (b) below.
From the image, we see that the period, or time
to complete one cycle, is 0.80 s.
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• Homework Practice page 375 and Formative
  Assessment pg. 371

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• Proof of Earths rotation?
• Foucault Pendulum

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Mass-Spring Systems

• The period of a mass-spring system depends on
  mass and spring constant.
• T 2? m/k

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Book example
The body of a 1275 kg car is supported on a frame
by four springs. Two people riding in the car
have a combined mass of 153 kg. When driven over
a pothole in the road, the frame vibrates with a
period of 0.840 s. For the first few seconds,
the vibration approximates simple harmonic
motion. Find the spring constant of a single
spring.
mtotal 1275kg 153
357kg/wheel 4 wheels 4
m car1275 kg mpeople 153 kg 4
wheels T0.840s k?

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• T 2? m/k
• Square both sidesand solve for k
• k4 ?2m/T2
• k4?2357kg/(0.84s)2
• k2.00X104N/m

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Practice and Formative Assessment pg. 377
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Example 1. A mass of 0.50 kg is attached to a
spring and is set into vibration with a period
of 0.42 s. What is the spring constant of the
spring?
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• Example 2. A 0.60 kg mass attached to
• a vertical spring stretches the spring 0.25 m.
• Find the spring constant.
• The system is now placed on a horizontal
• surface and set to vibrate. Find the period.

A)
B)
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What is a wave ?

• A definition of a wave
• A wave is a traveling disturbance that transports
  energy but not matter.
• Examples
• Sound waves (air moves back forth)
• Water waves (water moves up down)
• Light waves (do not require a medium)
• Mechanical waves require a medium
• Electromagnetic waves (like light) do not

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Measuring Waves

• Frequency (f) number of cycles per second
• Period time in seconds required for one
  complete cycle of motion
• Velocity(v) speed of wave in m/s
• Formulas
• v ?/T
• f 1/T
• v f ?

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Wave Properties

• Wavelength The distance ? between identical
  points on the wave.

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Example 1. A transverse wave has a wavelength of
0.80 m and a period of 0.25 s. Find the speed of
the wave.
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Example 2. A transverse wave has a wavelength of
40 cm and a frequency of 30.0 Hz. Find the speed
of the wave.
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Example 3. An astronaut broadcasts radio waves
with a speed (c) of 3.00 x 108 m/s and a
frequency (f) of 92.0 MHz. Calculate the
wavelength of these waves.
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Example 4. A certain laser emits light of
wavelength 633 nm. What is the Frequency of this
light in a vacuum?
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• Homework pg. 383 practice and 384 formative
  assessment

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Wave Types

• Pulse wave single, non-periodic disturbance.
• Periodic wave a wave whose source is some kind
  of periodic motion.

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• Transverse The medium oscillates perpendicular
  to the direction the wave is moving.
• Water (more or less)
• String waves

• Longitudinal The medium oscillates in the same
  direction as the wave is moving.
• Sound

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Wave Interactions

• Because mechanical waves are not matter, but are
  displacements of matter, they can occupy the same
  space at the same time.
• Superposition combination of two overlapping
  waves

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Wave Interactions

• Interference interaction of waves as a result
  of them passing through each other.

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• When the displacements are in the same direction,
  this produces Constructive Interference (see
  figure 4.3, pg. 386)
• The resultant amplitude of the resultant wave is
  the sum of the amplitudes of the individual
  waves.
• Each wave maintains its own characteristics
  after the interference

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• When the displacements are in the opposite
  direction, this produces Destructive Interference
  (see figure 4.4, pg. 387)
• The resultant amplitude of the resultant wave is
  the difference between the two pulses.
• If the difference is zero, then it is called
  Complete Destructive Interference (fig 4.5)
• Again, each wave maintains its own
  characteristics after the interference

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What is the resultant wave?
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Wave Motion
Waves and springs animation.
Wave pulse traveling down string, striking
fixed boundary.
Wave pulse traveling down string, striking free
boundary.
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Reflection

• At a free boundary (fig 4.6a pg. 388) waves are
  reflected.
• At a fixed boundary (fig 4.6b pg. 388) waves are
  reflected and inverted.

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Fixed Boundary
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Free Boundary
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Standing Waves

• A standing wave is produced when two waves with
  the same amplitude, wavelength and frequency
  travel in opposite directions and interfere (fig.
  4.7, pg. 389).
• Node point on a standing wave that always
  undergoes complete destructive interference and
  therefore is stationary.
• Antinode a point in a standing wave half-way
  between two nodes at which the largest amplitude
  occurs.

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If both ends of the string are fixed, a standing
wave results.
If the string is infinitely long and the
up-and-down pulses repeat, we get a traveling
wave.
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• Only certain frequencies will produce standing
  wave patterns.
• The ends have to be nodes, so this includes a
  wavelength of 2L (twice the length of the string
  or tube where the vibration is taking place), L,
  and 2/3 L. (see figure 4.8, pg. 390)

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Formative Assessment pg. 390
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Review Problems