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Lecture Outline
Chapter 13
College Physics, 7th Edition
Wilson / Buffa / Lou
© 2010 Pearson Education, Inc.
Chapter 13Vibrations and Waves
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Units of Chapter 13
Simple Harmonic Motion
Equations of Motion
Wave Motion
Wave Properties
Standing Waves and Resonance
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13.1 Simple Harmonic Motion
A mass on a spring is an example of periodic motion, in the absence of friction. It oscillates between its release point and a point equally far from equilibrium. This is called simple harmonic motion, as the restoring force has the simplest possible form:
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13.1 Simple Harmonic MotionDisplacement (x) is the directed distance of the object from equilibrium.
Amplitude (A) is the maximum displacement.
Period (T) is the time for one full cycle.
Frequency (f) is the number of full cycles per second.
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13.1 Simple Harmonic Motion
SI unit of frequency: hertz, Hz
1 Hz = 1 cycle/second
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13.1 Simple Harmonic Motion
Potential energy of a spring:
Total mechanical energy is conserved here; it is easiest to calculate the total energy at the endpoints of motion, where the energy is all potential energy.
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13.1 Simple Harmonic Motion
The total energy of an object in simple harmonic motion is directly proportional to the square of the amplitude of the object’s displacement.
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13.1 Simple Harmonic Motion
This allows us to calculate the velocity as a function of position:
and the maximum velocity (at x = 0):
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13.1 Simple Harmonic MotionThe energy varies from being completely kinetic to completely potential, and back again.
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13.2 Equations of MotionAn equation of motion gives the position of an object as a function of time.
Simple harmonic motion can be represented as a component of uniform circular motion:
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13.2 Equations of Motion
The equation of motion for the oscillating object is given by:
Here, the only possibility is that y = 0 at t = 0. More likely, we would want y = A at t = 0; that is, at t = 0 the object would have its maximum displacement. In that case,
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13.2 Equations of Motion
Relating the angular frequency to the period of motion gives, for y0 = 0,
The period of motion is given by:
and the frequency and angular frequency:
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13.2 Equations of Motion
This figure shows the displacement as a function of time.
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13.2 Equations of Motion
These graphs show the form of the equation of motion for different initial conditions.
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13.2 Equations of Motion
The velocity of an object in simple harmonic motion is:
and the acceleration:
Note that the acceleration is not constant—the equations of motion for constant acceleration cannot be used here.
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13.2 Equations of MotionIn the real world, friction will cause an oscillating object to slow down and eventually stop. This is called damped harmonic motion.
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13.3 Wave Motion
A wave pulse is a disturbance that propagates through a medium. It transfers energy without transferring matter; the energy is a combination of kinetic and potential energy.
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13.3 Wave Motion
A harmonic disturbance can set up a sinusoidal wave. The distance from crest to crest, or trough to trough, is called the wavelength, λ.
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13.3 Wave Motion
Such a wave will have a sinusoidal form in both time and space.
Amplitude: maximum displacement
Wavelength: distance between points having the same phase
Frequency: number of waves passing per second
Period: time for one complete wave to pass
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13.3 Wave Motion
Relationship between wave speed, wavelength, period, and frequency:
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13.3 Wave Motion
Waves may be either transverse (displacement perpendicular to direction of propagation) or longitudinal (displacement parallel to direction of propagation). Both types are illustrated here.
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13.3 Wave Motion
Water waves may appear to be transverse, but they are actually a combination of transverse and longitudinal motion.
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13.4 Wave Properties
When two or more waves travel through the same medium at the same time, they interfere in a process called superposition.
At any time, the combined waveform of two or more interfering waves is given by the sum of the displacements of the individual waves at each point in the medium.
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13.4 Wave Properties
This is an illustration of interference, using the principle of superposition. The displacement of any point on the rope is the sum of the individual displacements:
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13.4 Wave PropertiesIf the combined wave is larger than the individual ones, the interference is constructive; if smaller, it is destructive.
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13.4 Wave PropertiesDestructive interference is used in noise-cancelling technology.
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13.4 Wave PropertiesWhether or not a wave is inverted upon reflection depends on whether the end is free to move or not.
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13.4 Wave Properties
When a wave enters a new medium, its speed usually changes, as the properties of the new medium are different. The direction of propagation changes also; this is called refraction.
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13.4 Wave Properties
If the speed of the wave depends on its wavelength, it exhibits dispersion. The rainbow of light from a prism is an example of dispersion.
Diffraction occurs when a wave passes through an opening that is comparable in size to the wavelength; the waves will “bend” around the edges of the opening.
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13.5 Standing Waves and Resonance
On a rope with one fixed end, it is possible to set up waves that do not travel; they simply vibrate in place. These are called standing waves.
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13.5 Standing Waves and Resonance
Some points on the wave remain stationary all the time; these are called nodes. Others have the maximum displacement; these are called antinodes. Adjacent nodes are separated by half a wavelength, as are adjacent antinodes.
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13.5 Standing Waves and Resonance
When an integral number of half-wavelengths fit on the rope, the frequency is called the resonant frequency.
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13.5 Standing Waves and Resonance
Natural frequencies for a stretched string:
The wave speed is given by
where FT is the tension and μ is the mass per unit length.
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13.5 Standing Waves and Resonance
Natural wavelengths can be varied by varying the length of a string, such as in a piano or harp; varying the mass per unit length of a string, as in a guitar; or varying the tension, which is done for fine tuning.
Driving a system at its natural frequency produces resonance; the amplitude at resonance is limited only by damping and by the strength of the materials.
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Summary of Chapter 13
Simple harmonic motion requires a restoring force proportional to the displacement.
The frequency is the inverse of the period.
The total energy is proportional to the square of the amplitude.
The equations of motion are sinusoidal.
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Summary of Chapter 13
Velocity and period of a mass on a spring:
Velocity and acceleration of a mass in simple harmonic motion:
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Summary of Chapter 13
A wave is a disturbance in space and time; wave motion transfers energy.
The combined amplitude of two or more interfering waves is the sum of their individual amplitudes at each point.
Standing waves may be produced on a string with fixed ends. Natural frequencies:
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