Piri Reis University 2011-2012/ Physics -I
Physics for Scientists & Engineers, with Modern Physics, 4th edition Giancoli
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Chapter 14-15-16 Oscillations, Wave Motion and Sound
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Units of Chapter 14 •Simple Harmonic Motion •Energy in the Simple Harmonic Oscillator •The Period and Sinusoidal Nature of SHM
•The Simple Pendulum •Damped Harmonic Motion •Forced Vibrations; Resonance
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14-2 Simple Harmonic Motion 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.
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14-2 Simple Harmonic Motion 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:
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14-2 Simple Harmonic Motion • 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
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14-2 Simple Harmonic Motion • 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
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14-2 Simple Harmonic Motion
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.
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14-2 Simple Harmonic Motion
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.
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14-3 Energy in the Simple Harmonic Oscillator We already know that the potential energy of a spring is given by:
The total mechanical energy is then:
The total mechanical energy will be conserved, as we are assuming the system is frictionless.
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14-3 Energy in the Simple Harmonic Oscillator 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:
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14-3 Energy in the Simple Harmonic Oscillator The total energy is, therefore And we can write:
This can be solved for the velocity as a function of position:
where
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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.
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The Period and Sinusoidal Nature of SHM Therefore, we can use the period and frequency of a particle moving in a circle to find the period and frequency:
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The Period and Sinusoidal Nature of SHM We can similarly find the position as a function of time:
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The Period and Sinusoidal Nature 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.
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The Period and Sinusoidal Nature of SHM The velocity and acceleration can be calculated as functions of time; the results are below, and are plotted at left.
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14-5 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.
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14-5 The Simple Pendulum In order to be in SHM, the restoring force must be proportional to the negative of the displacement. Here we have:
which is proportional to sin θ and not to θ itself.
However, if the angle is small, sin θ ≈ θ.
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14-5 The Simple Pendulum Therefore, for small angles, we have:
where The period and frequency are:
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14-5 The Simple Pendulum
So, as long as the cord can be considered massless and the amplitude is small, the period does not depend on the mass.
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14-7 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.
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14-7 Damped Harmonic Motion 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.
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14-7 Damped Harmonic Motion 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.
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14-8 Forced Vibrations; Resonance 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.
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14-8 Forced Vibrations; 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.
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HOMEWORK Giancoli, Chapter 14
4, 6, 11, 13, 17, 22, 24, 25, 28, 32
References o “Physics For Scientists &Engineers with Modern Physics” Giancoli 4th edition, Pearson International Edition
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Units of Chapter 15 •Wave Motion •Types of Waves: Transverse and Longitudinal •Energy Transported by Waves •Intensity Related to Amplitude and Frequency
•Reflection and Transmission of Waves •Interference; Principle of Superposition •Standing Waves; Resonance •Refraction •Diffraction
•Mathematical Representation of a Traveling Wave 28
15-1 Wave Motion
A wave travels along its medium, but the individual particles just move up and down.
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15-1 Wave Motion 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.
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15-1 Wave Motion Wave characteristics: • Amplitude, A
• Wavelength, λ • Frequency f and period T • Wave velocity
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15-2 Types of Waves: Transverse and Longitudinal
The motion of particles in a wave can either be perpendicular to the wave direction (transverse) or parallel to it (longitudinal).
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15-2 Types of Waves: Transverse and Longitudinal Sound waves are longitudinal waves:
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15-2 Types of Waves: Transverse and Longitudinal 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.
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15-3 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:
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15-3 Energy Transported by Waves 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:
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Intensity Related to Amplitude and Frequency 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: (11-17) Therefore, the intensity is proportional to the square of the frequency and to the square of the amplitude.
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15-4 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.
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15-4 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:
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15-7 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. 40
15- 7 Reflection and Transmission of Waves
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.
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15-7 Reflection and Transmission of Waves 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.
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15-7 Reflection and Transmission of Waves The law of reflection: the angle of incidence equals the angle of reflection.
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15-8 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.
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15-8 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.
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15-9 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.
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15-9 Standing Waves; Resonance
The frequencies of the standing waves on a particular string are called resonant frequencies. They are also referred to as the fundamental and harmonics.
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15-9 Standing Waves; Resonance The wavelengths and frequencies of standing waves are:
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15-10 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:
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15-10 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.
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15-11 Diffraction
When waves encounter an obstacle, they bend around it, leaving a “shadow region.” This is called diffraction.
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15-11 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).
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Chapter 16 Sound
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Units of Chapter 16 •Characteristics of Sound •Intensity of Sound: Decibels •The Ear and Its Response; Loudness •Sources of Sound: Vibrating Strings and Air Columns
•Quality of Sound, and Noise; Superposition •Interference of Sound Waves; Beats •Doppler Effect •Shock Waves and the Sonic Boom •Applications: Sonar, Ultrasound, and Medical Imaging 54
16-1 Characteristics of Sound Sound can travel through any kind of matter, but not through a vacuum.
The speed of sound is different in different materials; in general, it is slowest in gases, faster in liquids, and fastest in solids. The speed depends somewhat on temperature, especially for gases.
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16-1 Characteristics of Sound Loudness: related to intensity of the sound wave Pitch: related to frequency.
Audible range: about 20 Hz to 20,000 Hz; upper limit decreases with age Ultrasound: above 20,000 Hz; see ultrasonic camera focusing below Infrasound: below 20 Hz
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16-2 Intensity of Sound: Decibels The intensity of a wave is the energy transported per unit time across a unit area.
The human ear can detect sounds with an intensity as low as 10-12 W/m2 and as high as 1 W/m2. Perceived loudness, however, is not proportional to the intensity.
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16-2 Intensity of Sound: Decibels The loudness of a sound is much more closely related to the logarithm of the intensity.
Sound level is measured in decibels (dB) and is defined:
I0 is taken to be the threshold of hearing:
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16-2 Intensity of Sound: Decibels An increase in sound level of 3 dB, which is a doubling in intensity, is a very small change in loudness. In open areas, the intensity of sound diminishes with distance:
However, in enclosed spaces this is complicated by reflections, and if sound travels through air the higher frequencies get preferentially absorbed.
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12-3 The Ear and Its Response; Loudness
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12-3 The Ear and Its Response; Loudness
Outer ear: sound waves travel down the ear canal to the eardrum, which vibrates in response
Middle ear: hammer, anvil, and stirrup transfer vibrations to inner ear Inner ear: cochlea transforms vibrational energy to electrical energy and sends signals to the brain
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16-3 The Ear and its Response; Loudness The ear’s sensitivity varies with frequency. These curves translate the intensity into sound level at different frequencies.
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16-4 Sources of Sound: Vibrating Strings and Air Columns Musical instruments produce sounds in various ways – vibrating strings, vibrating membranes, vibrating metal or wood shapes, vibrating air columns.
The vibration may be started by plucking, striking, bowing, or blowing. The vibrations are transmitted to the air and then to our ears.
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16-4 Sources of Sound: Vibrating Strings and Air Columns The strings on a guitar can be effectively shortened by fingering, raising the fundamental pitch.
The pitch of a string of a given length can also be altered by using a string of different density.
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16-4 Sources of Sound: Vibrating Strings and Air Columns A piano uses both methods to cover its more than seven-octave range – the lower strings (at bottom) are both much longer and much thicker than the higher ones.
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16-4 Sources of Sound: Vibrating Strings and Air Columns Wind instruments create sound through standing waves in a column of air.
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16-4 Sources of Sound: Vibrating Strings and Air Columns A tube open at both ends (most wind instruments) has pressure nodes, and therefore displacement antinodes, at the ends.
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16-4 Sources of Sound: Vibrating Strings and Air Columns A tube closed at one end (some organ pipes) has a displacement node (and pressure antinode) at the closed end.
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16-5 Quality of Sound, and Noise; Superposition So why does a trumpet sound different from a flute? The answer lies in overtones – which ones are present, and how strong they are, makes a big difference. The plot below shows frequency spectra for a clarinet, a piano, and a violin. The differences in overtone strength are apparent.
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16-6 Interference of Sound Waves; Beats Sound waves interfere in the same way that other waves do in space.
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16-6 Interference of Sound Waves; Beats Waves can also interfere in time, causing a phenomenon called beats. Beats are the slow “envelope” around two waves that are relatively close in frequency.
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16-7 Doppler Effect The Doppler effect occurs when a source of sound is moving with respect to an observer.
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16-7 Doppler Effect
As can be seen in the previous image, a source moving toward an observer has a higher frequency and shorter wavelength; the opposite is true when a source is moving away from an observer.
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16-7 Doppler Effect
If we can figure out what the change in the wavelength is, we also know the change in the frequency.
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16-7 Doppler Effect The change in the wavelength is given by:
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16-7 Doppler Effect And the change in the frequency:
If the source is moving away from the observer:
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16-7 Doppler Effect If the observer is moving with respect to the source, things are a bit different. The wavelength remains the same, but the wave speed is different for the observer.
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16-7 Doppler Effect We find, for an observer moving towards a stationary source:
And if it is moving away:
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16-8 Shock Waves and the Sonic Boom If a source is moving faster than the wave speed in a medium, waves cannot keep up and a shock wave is formed.
The angle of the cone is:
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16-8 Shock Waves and the Sonic Boom Shock waves are analogous to the bow waves produced by a boat going faster than the wave speed in water.
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16-8 Shock Waves and the Sonic Boom Aircraft exceeding the speed of sound in air will produce two sonic booms, one from the front and one from the tail.
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16-9 Applications: Sonar, Ultrasound, and Medical Imaging Sonar is used to locate objects underwater by measuring the time it takes a sound pulse to reflect back to the receiver. Similar techniques can be used to learn about the internal structure of the Earth. Sonar usually uses ultrasound waves, as the shorter wavelengths are less likely to be diffracted by obstacles.
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16-9 Applications: Sonar, Ultrasound, and Medical Imaging Ultrasound is also used for medical imaging. Repeated traces are made as the transducer is moved, and a complete picture is built.
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16-9 Applications: Sonar, Ultrasound, and Medical Imaging Ordinary ultrasound gives a good picture; high-resolution ultrasound is excellent.
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Summary of Chapter 16 • Sound is a longitudinal wave in a medium. • The pitch of the sound depends on the frequency. • The loudness of the sound depends on the intensity and also on the sensitivity of the ear. • The strings on stringed instruments produce a fundamental tone whose wavelength is twice the length of the string; there are also various harmonics present.
•Wind instruments have a vibrating column of air when played. If the tube is open, the fundamental is twice its length; if it is closed the fundamental is four times the tube length. • Sound waves exhibit interference; if two sounds are at slightly different frequencies they produce beats. • The Doppler effect is the shift in frequency of a sound due to motion of the source or the observer. 85
HOMEWORK Giancoli, Chapter 16
3, 6, 10, 16, 18, 69 References o “Physics For Scientists &Engineers with Modern Physics” Giancoli 4th edition, Pearson International Edition
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