Vibrations and Waves

1
Chapter 13

• Vibrations and Waves

Conceptual questions 2,6,8,17,19 Quick quizzes
2,6,7 Problems 12,34,50
2
Hookes Law

• Fs - k x
• Fs is the spring force
• k is the spring constant, it measures the
  stiffness of the spring
• x is the displacement of the object from its
  equilibrium position
• The negative sign indicates that the force is
  always directed opposite to the displacement

3
Simple Harmonic Motion

• Newtons second law will relate force and
  acceleration
• The force is given by Hookes Law
• F - k x m a
• a -kx / m
• The acceleration is a function of position
• Acceleration is not constant and therefore the
  uniformly accelerated motion equation cannot be
  applied

4
Elastic Potential Energy

• The energy stored in a stretched or compressed
  spring or other elastic material is called
  elastic potential energy
• PEs ½kx2
• The energy is stored only when the spring is
  stretched or compressed
• Conservation of energy
• (KEPEgPEs)i (KEPEgPEs)f

5
Energy in a Spring Mass System

• A block sliding on a frictionless system collides
  with a light spring
• The block attaches to the spring
• Spring is compressed
• The spring decompresses and sends the mass with
  the initial speed

6
Problem 13-12

• A 1.50-kg block at rest on a tabletop is
  attached to a horizontal spring having constant
  19.6 N/m as in Figure P13.12. The spring is
  initially unstretched. A constant 20.0-N
  horizontal force is applied to the object causing
  the spring to stretch. Determine the speed of the
  block after it has moved 0.300 m from equilibrium
  if the surface between the block and tabletop is
  frictionless.

7
Velocity as a Function of Position

• From Conservation of Energy
• we find speed
• Speed is a maximum at x 0
• Speed is zero at x A
• The indicates the object can be traveling in
  either direction

8
Simple Harmonic Motion and Uniform Circular Motion

• A ball is attached to the rim of a turntable of
  radius A
• The focus is on the shadow that the ball casts on
  the screen
• When the turntable rotates with a constant
  angular speed, the shadow moves in simple
  harmonic motion

9
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10
Period and Frequency from Circular Motion

• Period
• Time required for an object to complete one cycle
  
• Frequency
• Units are cycles/second or Hertz, Hz

11
Angular Frequency

• The angular frequency is related to the frequency

12
Motion as a Function of Time

• x A cos (2pt)
• x is the position at time t
• x varies between A and -A

13
Graphical Representation of Motion

• When x is a maximum
• the velocity is a minimum (zero)
• acceleration is a maximum in the negative
  direction

14
Quick Quiz 13.2

• Which of the following combination of quantities
  cannot be in the same direction for a simple
  harmonic oscillator?
• (a) position and velocity,
• (b) velocity and acceleration,
• (c) position and acceleration.

15
Quick Quiz 13.6

• If the amplitude of a system moving in simple
  harmonic motion is doubled, which of the
  following quantities does not change?
• (a) total energy,
• (b) maximum speed,
• (c) maximum acceleration,
• (d) period.

16
Verification of Sinusoidal Nature

• This experiment shows the sinusoidal nature of
  simple harmonic motion
• The spring mass system oscillates in simple
  harmonic motion
• The attached pen traces out the sinusoidal motion

17
Simple Pendulum

• The simple pendulum is an example of simple
  harmonic motion
• The force is the component of the weight tangent
  to the path of motion
• F - m g sin ?

18
Simple Pendulum, cont

• In general, the motion of a pendulum is not
  simple harmonic
• However, for small angles, it becomes simple
  harmonic
• In general, angles lt 15 are small enough
• sin ? ?
• F - m g ?

19
Period of Simple Pendulum

• This shows that the period is independent of of
  the amplitude
• The period depends on the length of the pendulum
  and the acceleration of gravity at the location
  of the pendulum

20
Quick Quiz 13.7

• A simple pendulum is suspended from the ceiling o
  a stationary elevator, and the period is
  measured. When the elevator accelerates upward,
  what happens to the period? (a) increases, (b)
  decreases, or (c) remains the same.
• If the elevator moves with constant velocity,
  what happens to the period? (a) increases, (b)
  decreases, or (c) remains the same.

21
Question

• The period of a simple pendulum is measured on
  Earth to be T. If the same pendulum is taken to
  the Moon, its period on the Moons surface will
  be (a) less than T,
• (b) greater than T,
• (c) equal to T ?

22
Problem 13-34

• A simple pendulum is 5.00 m long. (a) What is
  the period of simple harmonic motion for this
  pendulum if it is located in an elevator
  accelerating upward at 5.00 m/s2? (b) What is its
  period if the elevator is accelerating downward
  at 5.00 m/s2? (c) What is the period of simple
  harmonic motion for this pendulum if it is placed
  in a truck that is accelerating horizontally at
  5.00 m/s2.

23
Simple Pendulum Compared to a Spring-Mass System
24
Damped Oscillations

• Friction reduces the total energy of the system
  and the oscillation is said to be damped

25
Conceptual questions

• 2. If a spring is cut in half, what happens to
  its spring constant?
• 6. If an object-spring system is hung vertically
  and set into oscillation, why does the motion
  eventually stop?
• 8. If a pendulum clock keeps perfect time at the
  base of a mountain, will it also keep perfect
  time when moved to the top of the mountain?

26
Wave Motion

• A wave is the motion of a disturbance
• All waves carry energy and momentum
• Mechanical waves require
• A source
• A medium that can carry the wave
• Some physical mechanism though which energy
  transforms from one form into another

27
Types of Waves -- Transverse

• In a transverse wave, each element that is
  disturbed moves perpendicularly to the wave motion

28
Types of Waves -- Longitudinal

• In a longitudinal wave, the elements of the
  medium undergo displacements parallel to the
  motion of the wave

29
Waveform A Picture of a Wave

• The red curve is a snapshot of the wave at some
  instant in time
• The blue curve is later in time
• A is a crest of the wave
• B is a trough of the wave

30
Longitudinal Wave Represented as a Sine Curve

• A longitudinal wave can also be represented as a
  sine curve
• Compressions correspond to crests and stretches
  correspond to troughs

31
Description of a Wave

• Amplitude is the maximum displacement of string
  above the equilibrium position
• Wavelength, ?, is the distance between two
  successive points that behave identically

yA sin(2pft)
v ?/Tfl
32
Speed of a Wave

• v ?
• Is derived from the basic speed equation of
  distance/time
• This is a general equation that can be applied to
  many types of waves

33
Speed of a Wave on a String

• The speed of a wave on a spring stretched under
  some tension, F
• The speed depends only upon the properties of the
  medium through which the disturbance travels

34
Problem 13-50

• The elastic limit of a piece of steel wire is
  2.70 x 109 Pa. What is the maximum speed at which
  transverse wave pulses can propagate along this
  wire without exceeding this stress? (The density
  of steel is 7.86 x 103 kg/m3).

35
Interference of Waves

• Waves obey the Superposition Principle
• If two or more traveling waves are moving through
  a medium, the resulting wave is found by adding
  together the displacements of the individual
  waves point by point

36
Constructive Interference

• Two waves, a and b, have the same frequency and
  amplitude
• The combined wave, c, has the same frequency and
  a greater amplitude

37
Destructive Interference

• Two waves, a and b, have the same amplitude and
  frequency
• They are 180 out of phase
• When they combine, the waveforms cancel

38
Reflected Wave Free End

• When a traveling wave reaches a boundary, all or
  part of it is reflected
• When reflected from a free end, the pulse is not
  inverted

39
Conceptual questions

• 17. Explain why the kinetic and potential
  energies of an object-spring system can never be
  negative.
• 19. By what factor would you have to multiply the
  tension in a stretched spring in order to double
  the wave speed?