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### UNIT FOURWave Motion and Optics

PHYSICS OF TECHNOLOGYSpring 2009 Assignment Sheet

*Homework Handout

Wave motion describes phenomena ranging from the familiar...

Ocean waves...

waves...

(gravity waves).

Restoring Forces and Oscillations

- A restoring force is a force that exerts a push or a pull back towards equilibrium.
- A restoring force that increases in direct proportion to the distance from equilibrium results in simple harmonic motion.

Springs and Simple Harmonic Motion

- Simple harmonic motion occurs when the energy of a system repeatedly changes from potential energy to kinetic energy and back again.

Energy added by doing work to stretch the spring is transformed back and forth between potential energy and kinetic energy.

The horizontal position x of the mass on the spring is plotted against time as the mass moves back and forth.

Oscillatory Motion

- The period Tis the time taken for one complete cycle.
- The frequency fis the number of cycles per unit time. F=1/T
- The amplitudeis the maximum distance from equilibrium.

X(t) = A sin (2π f t)

Energy and Oscillations

Why does a swinging pendant return to the same point after each swing?

Energy and Oscillations

The force does work to move the ball. This increases the ball’s energy, affecting its motion.

+

+

+

+

+

+

+

+

Compressionand Oscillation on an Atomic ScaleBonds between atoms in a compressed solid can be treated as compressed springs.

Ultimately the forces come from electrostatic interactions between electrons and protons (and a little quantum mechanics).

Fspring=-k Δx

Water seems to move toward the shore, but no water accumulates on the beach.

What Are Waves?

What is actually moving?

Do the waves carry energy?

Wave Pulses and Periodic Waves

A Slinky is ideal for studying simple waves.

Wave Pulses and Periodic Waves

- If a Slinky is laid out on a smooth table with one end held motionless, you can easily produce a single traveling pulse:
- With the Slinky slightly stretched, move the free end back and forth once along the axis of the Slinky.
- You will see a disturbance (the wave pulse) move from the free end of the Slinky to the fixed end.
- What is actually moving?
- The pulse moves through the Slinky, and portions of the Slinky move as the pulse passes through it.
- After the pulse dies out, the Slinky is exactly where it was before the pulse began.

Wave Pulses and Periodic Waves

- Moving one end of the Slinky back and forth created a local compression where the rings of the spring are closer together than in the rest of the Slinky.
- This region of compression moves along the Slinky and constitutes the pulse.
- The wave or pulse moves through the medium (here, the Slinky), but the medium goes nowhere.
- What moves is a disturbance within the medium which may be a local compression, a sideways displacement (like a wave on a rope), etc.
- The speed of the pulse may depend on factors such as tension in the Slinky and the mass of the Slinky.

Wave Pulses and Energy

- Energy is transferred through the Slinky as the pulse travels.
- The work done in moving one end of the Slinky increases both the potential energy of the spring and the kinetic energy of individual loops.
- This region of higher energy then moves along the Slinky and reaches the opposite end.
- There, the energy could be used to ring a bell or perform other types of work.
- Energy carried by water waves does substantial work over time in eroding and shaping a shoreline.

Longitudinal Waves

- The pulse we have been discussing is a longitudinal wave: the displacement or disturbance in the medium is parallel to the direction of travel of the wave or pulse.
- Sound waves are

longitudinal.

Transverse Waves

- By moving your hand up and down, you could also produce a transverse wave, in which the displacement or disturbance is perpendicular to the direction the wave is traveling.
- Waves on a rope and

electromagnetic waves

are transverse.

- Polarization effects are

associated with

transverse waves but

not longitudinal waves.

- Water waves have

both longitudinal and

transverse properties.

Generating Longitudinal Waves

- If instead of moving your hand back and forth just once, you continue to produce pulses, you will send a series of longitudinal pulses down the Slinky.
- If equal time intervals separate the pulses, you produce a periodic wave.
- The time between pulses is the period T of the wave.
- The number of pulses or cycles per unit of time is the frequency f = 1/T.
- The distance between the same points on successive pulses is the wavelength .
- A pulse travels a distance of one wavelength in a time of one period.
- The speed is then the wavelength divided by the period:

A longitudinal wave traveling on a Slinky has a period of 0.25 s and a wavelength of 30 cm. What is the frequency of the wave?

Longitudinal Waves

- 0.25 Hz
- 0.30 Hz
- 0.83 Hz
- 1.2 Hz
- 4 Hz

A longitudinal wave traveling on a Slinky has a period of 0.25 s and a wavelength of 30 cm. What is the speed of the wave?

Longitudinal Waves

- 0.25 cm/s
- 0.30 cm/s
- 1 cm/s
- 7.5 cm/s
- 120 cm/s

Generating A Transverse Pulse

- A snapshot of a single transverse pulse moving along a rope is like a graph of the vertical displacement of the rope plotted against the horizontal position.

- At some later time the pulse will be farther down the rope at a different horizontal position.
- The shape remains basically the same.

Generating A Transverse Wave

- If you repeat a series of identical pulses at regular time intervals, you might produce a periodic wave such as shown.
- The wavelength is the distance covered by one complete cycle of the wave.
- This wave pattern moves to the right along the rope, retaining its shape.
- The shape depends on the exact motion of the hand or other oscillator generating the wave.
- When the leading edge of the wave reaches the fixed end of the rope, it will be reflected and start to move back to the left.
- The reflected wave will interfere with the wave still traveling to the right.

Generating A n Harmonic Wave

- If you move your hand up and down smoothly in simple harmonic motion, the displacement of this end of the rope will vary sinusoidally with time.
- The resulting periodic wave will also have a sinusoidal form.
- Such a wave is called a harmonic wave.
- The individual segments of rope tend to move with simple harmonic motion, because the restoring force pulling the rope back toward the center line is proportional to its distance from the center line.
- Any periodic wave can be represented as a sum of harmonic waves with different wavelengths and frequencies.
- The process of breaking a complex wave down into its simple harmonic components is called Fourier, or harmonic analysis.

A wave on a rope is shown below. What is the wavelength of this wave?

Wavelength

a) 1/6 m b) 1 m c) 2 m d) 3 m e) 6 m

In 6 m, the wave goes through 2 complete cycles.

The wavelength (length of one complete cycle) is (6 m)/2 = 3 m.

If the frequency of the wave is 2 Hz, what is the wave speed?

Frequency of a Wave

a) 1/6 m/s b) 2/3 m/s c) 2 m/s d) 3 m/s e) 6 m/s

- As the raised portion of a pulse approaches a given point on the rope, the tension in the rope acquires an upward component.
- The resulting upward force

causes this next segment

to accelerate upward, and

so on down the rope.

- The speed of the pulse

depends on how fast

succeeding segments can be

started moving (accelerated).

- By Newton’s second law, this is proportional to the force and inversely proportional to the mass of the segment:
- A larger tension produces a larger acceleration.
- The speed of the pulse will increase with the
- tension and decrease with the mass per unit
- length of the rope:

A rope has an overall length of 10 m and a total mass of 2 kg. The rope is stretched with a tension of 50 N. One end of the rope is fixed, and the other is moved up and down with a frequency of 4 Hz. What is the speed of waves on this rope?

Velocity of a Wave

a) 5.0 m/s b) 7.07 m/s c) 15.8 m/s d) 50 m/s e) 250 m/s

A rope has an overall length of 10 m and a total mass of 2 kg. The rope is stretched with a tension of 50 N. One end of the rope is fixed, and the other is moved up and down with a frequency of 4 Hz. What is the wavelength?

Wavelength of a Wave

a) 0.20 m b) 3.95 m c) 10 m d) 15.8 m e) 25 m/s

Interference and Standing Waves

- When a wave on a rope reaches the fixed end of the rope, it is reflected and travels in the opposite direction back toward your hand.
- If the wave is periodic, the reflected wave interferes with the incoming wave.
- The resulting pattern becomes more complex and confusing.
- This process, in which two or more waves combine, is called interference.

Superposition

- Imagine a rope consisting of two identical segments spliced together to form a single rope of the same mass per unit length as the two original segments.
- Identical waves traveling on

the two identical segments

will combine to form a larger

wave on the single joined

rope.

- At all points, the height of the

individual waves will add

together to form a wave with

the same frequency and

wavelength but twice the

height of the initial two waves.

- Principle of Superposition:

When two or more waves combine, the resulting disturbance or displacement is equal to the sum of the individual disturbances.

Interference

- When the two waves are moving the same way at the same time, they are in phase.
- The resulting combined wave will be larger (have a greater height).
- If one wave is moving

upward when the other

wave is moving downward,

the two waves are

completely out of phase.

- If the two waves have the

same height, the resulting

combined displacement will

be zero.

- No wave is propagated beyond the junction.
- The result of adding two waves together depends on their phases as well as on their height or amplitude.
- When waves are in phase, we have constructive interference.
- When waves are out of phase, we have destructive interference.

Interference

- When two or more waves are traveling in the same direction, the difference in phase determines whether the interference will be constructive, destructive, or somewhere in between.
- When two waves are traveling in opposite directions, such as when a wave is reflected back on itself, the principle of superposition can be applied at different points on the string.
- At point A, the two waves cancel each other at all times.
- At this point, the string will not oscillate at all; this is called a node.

- At point B, both waves will be in phase at all times.
- The two waves always add, producing a displacement twice that of each wave by itself.
- This is called an antinode.

Standing Waves

- This pattern of oscillation is called a standing wave.
- The waves traveling in opposite directions interfere in a way that produces a standing or fixed pattern.
- The distance between adjacent nodes or adjacent antinodes is half the wavelength of the original waves.
- At the antinodes, the string is oscillating with a large amplitude.
- At the nodes, it is

not moving at all.

- At points between

the nodes and

antinodes, the

amplitude has

intermediate

values.

Pitch

- Guitars, pianos, and other stringed instruments produce music using standing waves on strings.
- The frequency of the sound wave equals the string’s frequency of oscillation and is related to the musical pitch.
- A higher frequency represents a higher-pitched sound.

Harmonics

- For a string fixed at both ends, the simplest standing wave, the fundamental or first harmonic, has nodes at both ends and an antinode in the middle.

- The wavelength is determined by the length of the string.
- Since the distance between nodes is half the wavelength, the wavelength must be twice the length of the string.
- The wave speed is determined by the tension in the string and the mass per unit length of the string.
- The frequency can then be found using the relationship v=f:

Octaves

- A string with a longer length L will result in a lower frequency.
- The effective length can also be shortened by placing your finger firmly on the string,

producing a higher-pitched

tone.

- Other patterns of oscillation

may also be produced.

- The second harmonic has a node at the midpoint of the string, and a wavelength equal to L.
- This wavelength is half the fundamental, so its frequency is twice the fundamental.
- Musically, this pitch would be an octave above the fundamental.

A string with a longer length L will result a lower frequency.

- The effective length can also be shortened by placing your finger firmly on the string,

producing a higher-pitched

tone.

- Other patterns of oscillation

may also be produced.

- The third harmonic has four nodes (counting the ones at the ends) and three antinodes, and a wavelength equal to two-thirds L.
- The resulting frequency is three times the fundamental and 3/2 that of the second harmonic.
- Musically, this is called a fifth above the second harmonic.

A guitar string has a mass of 4 g, a length of 74 cm, and a tension of 400 N. These values produce a wave speed of 274 m/s. What is its fundamental frequency?

- 1.85 Hz
- 3.70 Hz
- 185 Hz
- 274 Hz
- 370 Hz

A guitar string has a mass of 4 g, a length of 74 cm, and a tension of 400 N. These values produce a wave speed of 274 m/s. What is the frequency of the second harmonic?

- 92.5 Hz
- 123 Hz
- 185 Hz
- 370 Hz
- 740 Hz

Sound Waves

- Sound waves may be generated in many ways in addition to an oscillating string.
- Since sound waves reach our ears, they must be able to travel through air.

The bass string has been plucked, producing a blur near the middle where the amplitude is greatest.

Sound Waves

- A sound wave consists of pressure variations in air.
- The diaphragm of a speaker oscillates back and forth, producing regions of higher pressure and lower pressure.
- These regions propagate through the air as variations in air pressure and density, forming a longitudinal sound wave.

Sound Waves

- We can show these pressure variations in a graph of pressure plotted against position.
- The factors that determine the speed of sound are related to how rapidly one air molecule transmits changes in velocity to nearby molecules to propagate the wave.
- In room temperature air, sound

waves travel with a speed of

340 m/s or 750 MPH.

- For gases other than air, the

speed also depends on the

masses of the molecules or

atoms.

- Sound waves can also travel

through liquids and solids, often

with higher speeds.

Sound Waves

- Interference phenomena such as standing waves can be observed in sound waves.
- Many musical instruments produce standing waves in a tube or pipe.
- If the tube is closed at one end, such as a bottle, there is a displacement node at the closed end.
- At the open end, there is a displacement antinode.
- The frequency of the standing

wave can be found from the

speed of sound in air and the

wavelength:

where the wavelength is

determined by the length of

the tube.

The standing-wave patterns for the first three harmonics for a tube open at one end and closed at the other are represented as follows:

- The first harmonic or fundamental has a wavelength four times longer than the length of the tube.
- The wavelength of the second harmonic is equal to four-thirds of the length of the tube.
- The wavelength of the third harmonic is equal to four-fifths of the length of the tube.
- etc.

The Doppler Effect

- A moving source of sound, such as a car horn, seems to change pitch depending on its motion relative to the listener.

- As a car passes a stationary observer, the horn’s pitch changes from a higher pitch to a lower pitch.

The Doppler Effect

- Comparing the wavefronts for a stationary car horn and for a moving car horn illustrates why the pitch changes.

- When the car is approaching the observer, the wavefronts reaching the observer are closer together.
- When the car is moving away from the observer, the wavefronts reaching the observer are farther apart.

The Physics of Music

- Why do certain combinations of notes (chords) sound better than others?
- Musical notes actually have a mix of higher harmonics along with the fundamental frequency.
- The relationships between these different frequencies explain why some notes sound harmonious together, and others do not.
- An analysis of the mixture of frequencies is called harmonic or frequency analysis.

When a guitar string is plucked in the usual position, the second and third harmonics often dominate the harmonic spectrum.

- f1 is the frequency of the first harmonic.
- Since the string is usually plucked right where the second harmonic (f2 = 2f1) has an antinode, the second harmonic is strongly stimulated.
- The body of the guitar

also determines which

harmonics will be

reinforced.

When a guitar string is plucked near the bridge, many higher harmonics are present in the harmonic spectrum.

- This results in a twangy sound.
- The pitch still sounds the same, but the tone quality depends on the mixture of harmonics present.
- Different instruments generate different harmonic mixes, so they have different

tones.

- A trumpet produces a

lot of higher harmonics

so it sounds “bright”

or “brassy”.

- A flute can produce a

tone dominated by the

fundamental frequency,

with almost none of the

higher harmonics, so

the tone sounds “pure”.

When discussing musical notes, we describe the difference in frequencies in terms of musical intervals.

- Musical scales and intervals are based upon the ratios between the higher harmonics in the notes.
- The frequency of the second harmonic is twice that of the first harmonic. This interval is an octave, and corresponds to the interval from one note C to the next higher C (the first and eighth notes in the scale).
- The frequency of the third harmonic is 3/2 that of the second harmonic. This interval is

a fifth, between the first

and fifth notes in the scale

(C and G).

- The frequency of the fourth

harmonic is 4/3 that of the

third harmonic. This

interval is a fourth,

between the first and

fourth notes (C and F).

Just tuning involves tuning an instrument so that in one key, all the intervals have simple frequency ratios.

- In a different key, the tuning may not sound correct.
- Equally-tempered tuning is a compromise so that the ratios are all approximately correct, but not perfect.
- The ratios between

adjacent half steps on

the scale are all identical,

so the scales sound

correct regardless of

what key you are in.

- The frequency of the fifth harmonic is 5/4 that of the fourth harmonic. This interval is a major third, between the first and third notes (C and E).

A C-major scale begins with do on middle C having a frequency of approximately 264 Hz. Assuming that they have been tuned to the perfect ratios for the intervals in question, what should the frequency be for sol (G)?

- 88 Hz
- 132 Hz
- 176 Hz
- 396 Hz
- 528 Hz

A C-major scale begins with do on middle C having a frequency of approximately 264 Hz. Assuming that they have been tuned to the perfect ratios for the intervals in question, what should the frequency be for fa (F)?

- 176 Hz
- 198 Hz
- 352 Hz
- 440 Hz
- 528 Hz

A C-major scale begins with do on middle C having a frequency of approximately 264 Hz. Assuming that they have been tuned to the perfect ratios for the intervals in question, what should the frequency be for do at the top of the scale (high C)?

- 88 Hz
- 198 Hz
- 352 Hz
- 440 Hz
- 528 Hz

Combinations of notes sound harmonious when the higher harmonics overlap.

- When two notes are too close in pitch, beats can produce a dissonant buzz.
- The two waves come in and out of phase as time progresses.

Alternating Current and Household Circuits

- The current we draw from a wall outlet is alternating current (ac) rather than direct current (dc).
- Direct current implies that the current flows in a single direction from the positive terminal of a battery or power supply to the negative terminal
- Alternating current continually reverses its direction -- it flows first in one direction, then in the other, then back again.
- In North America the ac goes through 60 cycles each second (60 Hz).

Alternating Current and Household Circuits

- The plot of electric current as a function of time for an alternating current is a sinusoidal curve.
- The average value of an ordinary alternating current is zero.
- The power dissipated in a resistance is proportional to the square of the current.
- The effective current or rms current is obtained by squaring the current, averaging this value over time, and taking the square root of the result.
- The effective current Ieff is 0.707 times the peak current Ipeak.

Alternating Current and Household Circuits

- If we plot the voltage across an electrical outlet as a function of time, we get another sinusoidal curve.
- The effective value of this voltage is typically between 110 and 120 volts in North America.
- The standard household power supplied in this country is 115 volts, 60 hertz ac.
- Household circuits are wired in parallel so that different appliances can be added to or removed from the circuit without affecting the voltage available.

A 60-W light bulb is designed to operate on 120 V ac. What is the effective current drawn by the bulb?

Alternating Current and Household Circuits

- 0.2 A
- 0. 5 A
- 2.0 A
- 72 A
- 7200 A

Alternating Current and Household Circuits

- Household circuits are wired in parallel so that different appliances can be added to or removed from the circuit without affecting the voltage available.
- As you add more appliances, the total current drawn increases, because the total effective resistance of the circuit decreases when resistances are added in parallel.
- Since too large a current could cause the wires to overheat, a fuse or circuit breaker in series with one leg of the circuit will disrupt the circuit if the current gets too large.
- Appliances with larger power requirements (stoves, clothes dryers, etc) are usually connected to a separate 220-V line.

Generators

- The flux changes continuously from a maximum value in one direction, to zero, to a maximum value in the opposite direction.
- The induced voltage depends on the rate of change of the flux.
- When the flux is

increasing the fastest,

the voltage is a

maximum; when the

flux is decreasing the

fastest, the voltage is

a maximum in the

other direction

(negative).

Magnets and the Magnetic Force

- We are generally more familiar with magnetic forces than with electrostatic forces.
- Like the gravitational force and the electrostatic force, this force acts even when the objects are not touching one another.
- Is there a relationship between electrical effects and magnetism?
- Maxwell discovered that the electrostatic force and the magnetic force are really just different aspects of one fundamental electromagnetic force.
- Our understanding of that relationship has led to numerous inventions such as electric motors, electric generators, transformers, etc.

James Clerk Maxwell

Maxwell Equations (1865) described all of E&M. They predict the existence of EM waves:

c is speed of EM waves (light)

Hetrz (1888) showed EM waves:

have speed c

have wavelength

have both E and M components

reflect

Refract

Interfere

diffract

Electromagnetic WavesWhat do light, radio waves, microwaves, and X rays have in common?

- They all can travel through empty space.
- They all travel at the same speed.
- They all have no mass.
- All the above are true.
- Only answers a and b are true.

These are all forms ofelectromagnetic waves.Although seemingly quite different, they share many properties, including a, b, and c.

Electromagnetic Waves

- An electromagnetic wave consists of time-varying electric and magnetic fields, in directions perpendicular to each other as well as to the direction the wave is traveling.

Antennas

- The electric and the magnetic fields can be produced by charged particles.
- An electric field surrounds any charged particle.
- A magnetic field surrounds moving charged particles.

- A rapidly alternating electric current in a wire generates magnetic fields whose direction and magnitude change with time.
- This changing magnetic field in turn produces a changing electric field.

Electromagnetic Waves

- Likewise, a changing electric field produces a magnetic field.
- Maxwell realized a wave involving these fields could propagate through space:
- A changing magnetic field produces a changing electric field, which produces a changing magnetic field, etc...
- Thus a transverse wave of associated changing electric and magnetic fields is produced.

Speed of Light

- Maxwell predicted the speed of electromagnetic waves in a vacuum using the Coulomb constant k in Coulomb’s law and the magnetic force constant k in Ampere’s law:
- This was equal to the known value for the speed of light!

Fizeau’s wheel for measuring the speed of light

There is a wide spectrum of frequencies and wavelengths of electromagnetic waves.

Electromagnetic Spectrum

- Different types of electromagnetic waves have different wavelengths and frequencies.
- Together they form the electromagnetic spectrum.

What is the frequency of radio waves with a wavelength of 10 m?

- Since they all travel at the speed of light c in a vacuum, their frequencies and wavelengths are related by: v = c = f

= 10 m f = v /

v = c = 3 x 108 m/s = (3 x 108 m/s) / 10 m

v = f = 3 x 107 Hz

Waves in different parts of the electromagnetic spectrum differ not only in wavelength and frequency but also in how they are generated and what materials they will travel through.

- Radio waves are generated by accelerated charges in an oscillating electrical circuit.
- X rays come from energy transitions of atomic electrons.
- Gamma rays originate inside an atomic nucleus.
- Infrared light is radiated by all warm bodies.
- Oscillating atoms within the molecules of the warm body serve as the antennas.
- X rays will pass through materials that are opaque to visible light.
- Radio waves will pass through walls that light cannot penetrate.

Different wavelengths of visible light are associated with different colors.

- Violet is about 3.8 x 10-7 m.
- Wavelengths shorter than the violet comprise ultraviolet light.
- Red is about 7.5 x 10-7 m.
- Wavelengths longer than the red comprise infrared light.
- In between, the colors are red, orange, yellow, green, blue, indigo, and violet.

Wavelength and Color

- How do we perceive ?
- What causes different objects to have ?
- Why is the sky ?

color

different colors

BLUE

Newton demonstrated that white light is a mixture of colors.

- He showed that white light from the sun, after being split into different colors by one prism, can be recombined by a second prism to form white light again.

Physics of Technology

Next Lab/Demo: Electric Circuits

Magnetism

Thursday 1:30-2:45

ESLC 46

Ch 13 and 14

Next Class: Friday 10:30-11:20

BUS 318 room

Read Ch 14

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