physics_workshop

# Physics Workshop Class

This is a separate 1-credit course that requires registering with the registrar's office

## Workshop sections

Course # CRN Day, Time Place Workshop leader Email
PH 299 62874 Th 12:00-13:50 XSB 183 Lih Louise Kuhlman lihkuhlman [at] gmail
PH 299 62872 Th 14:00-15:50 KHSE 112 Elizabeth Christine Baird bairdliz
PH 299 65497 Fr 12:00-13:50 NH 224 Andrew Hart Dempsey adempsey
PH 299 66243 Fr 15:00-16:50 SRTC 162 James C Salamanca jcsalama

# Workshop 3 (Apr 16,17)

## Doppler Effect and Electromagnetic Waves

### Equations and Relations WS.3

1. Speed, wavelength, and frequency of a wave
• $v=\lambda\,f$
2. Beats
• $f_\text{beat}=\big|\,f_1-f_2\big|$
3. Doppler Effect, moving observer:
• $f'=\left(1\pm\frac{u_o}{v}\right)f$
4. Doppler Effect, moving source:
• $f'=\big(\!\frac{1}{1\mp\frac{u_s}{v}}\!\big)\,f$
• $v$: speed of the wave
• $u_o$: relative speed of the observer
• $u_s$: relative speed of the source
5. Doppler Effect for electromagnetic waves:
• $f'=f\left(1\pm\frac{u}{c}\right)$
• $u$: relative speed between source and observer
6. Intensity as power per area:
• $I=\frac{P}{A}$
• $I=\frac{P}{4\pi r^2}$ (from a point source at a distance r)
7. Energy density (not to confuse with relative speed in Doppler effect!):
• $u=\frac{1}{2}\epsilon_0E^2+\frac{1}{2\mu_0}B^2$
8. Ratio of electric and magnetic field:
• $E=cB$
9. Intensity in terms of the energy density $u$ and speed of light $c$:
• $I=uc=\frac{1}{2}c\epsilon_0E^2+\frac{1}{2\mu_0}cB^2$
10. Momentum in terms of total energy $U$ and speed of light $c$:
• $p=\frac{U}{c}$
• $P_\text{av}=\frac{I_\text{av}}{c}$

### Problem 3.1

An auditorium has organ pipes at the front and at the rear of the hall. Two identical pipes, one at the front and one at the back, have fundamental frequencies of 264 Hz at 20$\,^\circ$C. During a performance, the organ pipes at the back of the hall are at 25$\,^\circ$C, while those at the front are still at 20$\,^\circ$C. What is the beat frequency when the two pipes sound simultaneously? (use: $v_{20\,^\circ{\text C}}=343\frac{\text m}{\text s}$, $\;\;v_{25\,^\circ{\text C}}=346\frac{\text m}{\text s}$, calculated with equation from thermodynamics: $v=\sqrt{\frac{\gamma RT}{M}}$, where M is the (average) molar mass, and $\gamma=\frac{c_p}{c_v}$ is the adiabatic constant).

Solution:

Questions:

• [Students, post your questions here]

### Problem 3.2

A source of sound waves of frequency 1.0 kHz is stationary. An observer is traveling at 0.50 times the speed of sound.

1. What is the observed frequency if the observer moves toward the source?
2. What is the observed frequency if the observer moves away from the source instead?

Solution:

Questions:

• [Students, post your questions here]

### Problem 3.3

A source of sound waves of frequency 1.0 kHz is traveling through the air at 0.50 times the speed of sound.

1. Find the frequency of the sound received by a stationary observer if the source moves towards her.
2. Repeat if the source moves away from her instead.

Solution:

Questions:

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### Problem 3.4

You drive in a your car at a speed of 50 km/h and ambulance approaches from behind at a speed of 80 km/h. When the ambulance is at rest its siren produces sound at a frequency of 1050 Hz.

1. What is the frequency of the siren observed by you?
2. What is the wavelength of sound reaching you?

Solution:

Questions:

• [Students, post your questions here]

### Question 3.1

The source and observer of a sound wave are both at rest with respect to the ground. The wind blows in the direction of source to observer. Is the observed frequency Doppler-shifted? Explain.

Questions:

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### Problem 3.5

A galaxy emits light at a wavelength of 656 nm. On earth the wavelength is measured to be 659.1 nm.

1. What is the speed of the galaxy relative to the earth?
2. Is the galaxy approaching or receding?

Solution:

Questions:

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### Questions 3.2-3.3

• What do cosmologist mean when the say the electromagnetic radiation from other galaxies is “red shifted”?
• Can you distinguish between the case of a moving observer or moving source in the case of sound and electromagnetic waves?

Questions:

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### Problem 3.6

When light having vibrations with angular frequency ranging from $2.7\times 10^{15}\frac{\text{rad}}{\text s}$ to $4.7\times 10^{15}\frac{\text{rad}}{\text s}$ strikes the retina of the eye, it stimulates the receptor cells there and is perceived as visible light. What are the limits of the period and frequency of this light?

Solution:

Questions:

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### Question 3.4

Microwave ovens, radio, radar, and x-rays utilize electromagnetic waves. Compare the energy, frequency and wavelengths of these waves to those of visible radiation.

### Problem 3.7

A lightning flash is seen in the sky and 8.2 s later the boom of the thunder is heard. The temperature of the air is 12$\,^\circ$C. (use $v_{12\,^\circ{\text C}}=338\frac{\text m}{\text s}$) How far away is the lightning strike?

Solution:

Questions:

• [Students, post your questions here]

### Questions 3.5-3.6

• During a thunderstorm, you can easily estimate your distance from a lightning strike. Count the number of seconds that elapse from when you see the flash of lightning to when you hear the thunder. The rule of thumb is that 5 seconds elapse for each mile distance. Verify that this rule of thumb is (approximately) correct. (One mile is 1.6 km and light travels at a speed of $c=3\times 10^8\frac{\text m}{\text s}$.)
• For an xyz coordinate system, if the E-vector is in the z direction, and the B-vector is in the x direction, what is the direction of propagation of the electromagnetic waves?

Questions:

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### Problem 3.8

The microwave background radiation leftover from the big bang has an average energy density of $4\times 10^{-14}\frac{\text J}{ {\text m}^3}$.

1. What is the rms and maximum value of the electric and magnetic component of this radiation?
2. Calculate the intensity of this radiation.

Solution:

1. the field values are:
• rms values
• $E_\text{rms}=\ldots\,\frac{\text V}{\text m}$
• $B_\text{rms}=\ldots\times 10^{-10}\,{\text T}$
• max values
• $E_\text{max}=\ldots\,\frac{\text V}{\text m}$
• $B_\text{max}=\ldots\times 10^{-10}\,{\text T}$
2. intensity
• $I=\ldots\,\frac{\text W}{\,{\text m}^2}$

Questions:

• [Students, post your questions here]

### Problem 3.9

The average intensity of the sunlight reaching the earth is $1390\frac{\text W}{ {\text m}^2}$.

1. What is the average radiation pressure due to the sunlight?
2. What is the maximum energy that a $5 \times 8\,$m solar panel could collect in 12 hours, if all sunlight is absorbed (the real efficiency of solar panels is much lower)?
3. Calculate the average force exerted by the light on the solar panel assuming it absorbs all incoming light.
4. Calculate the energy density of sunlight.

Solution:

Questions:

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### Questions 3.7-3.10

• Suppose you triple the magnitude of the magnetic field of an electromagnetic wave. By what factor does the electric component of the wave change? By what factor changes the intensity of the wave?
• In tennis a radar gun is often used to measure the speed of the ball. Describe how such a radar gun could work.
• If a spaceship uses the radiation pressure of the sun, should it use sails that are reflection or absorbing?
• If you move on a spaceship away from the sun, by how much does the intensity of the radiation decrease as you double the distance between you and the sun?

Questions:

• [Students, post your questions here]

# Workshop 2 (Apr 9,10)

## Waves on a String and Sound

### Equations and Relations WS.2

1. Speed, wavelength, and frequency of a wave
• $v=\lambda\,f$
2. Speed of a wave of a string:
• $v=\sqrt{\frac{F_T}{\mu}}$, where $F_T$ is the tension force, and $\mu=\frac{m}{L}$ is the mass per length (string)
3. Harmonic wave function
• $y(x,t)=A\cos\left(\frac{2\pi}{\lambda}x-\frac{2\pi}{T}t\right)$
• $k=\frac{2\pi}{\lambda}\;$ is the wavenumber, not to be confused with the spring constant
• and the angular frequency is $\omega=\frac{2\pi}{T}$
4. Intensity as power per area:
• $I=\frac{P}{A}$
• $I=\frac{P}{4\pi r^2}\;$ (point source at a distance r)
5. Intensity level $\beta$ of a sound wave:
• $\beta=10\log\big(\frac{I}{I_0}\big)$
• $I_0=10^{-12}\frac{\text W}{ {\text m}^2}$
6. Standing waves on a string:
• $f_n=nf_1=n\frac{v}{2L}$
• $\lambda_n=\frac{\lambda_1}{n}=\frac{2L}{n}\;\;$ where $n=1,2,3\ldots$ is the mode number
7. Vibrating columns of air – closed at one end
• $f_n=nf_1=n\frac{v}{4L}$
• $\lambda_n=\frac{\lambda_1}{n}=\frac{4L}{n}\;\;$ where $n=1,3,5\ldots$ is the mode number
8. Vibrating columns of air – open at both ends
• $f_n=nf_1=n\frac{v}{2L}$
• $\lambda_n=\frac{\lambda_1}{n}=\frac{2L}{n}\;\;$ where $n=1,2,3\ldots$ is the mode number
9. $L$ is the length of the tube or string

### Problem 2.1

When sound waves strike the eardrum, the membrane vibrates with the same frequency as the sound. The highest pitch that typical humans can hear has a period of 50.0 $\mu$s. What are the frequency and angular frequency of the vibrating eardrum for this sound?

Solution:

Questions:

• [Students, post your questions here]

### Problem 2.2

High-frequency sound waves (ultrasound) are used to probe the interior of the body, much as x-rays do. To detect small objects, such as tumors, a frequency of around 5.0 MHz is used. What are the period and angular frequency of the molecular vibrations caused by this pulse of sound?

Solution:

Questions:

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### Problem 2.3

A variable oscillator allows a laboratory student to adjust the frequency of a source to produce standing waves in a vibrating string. A 1.20-m length of string ($\mu$ = 0.400 g/m) is placed under a tension of 200 N. What frequency is necessary to produce three standing loops in the vibrating string? What is the fundamental frequency? What frequency will produce five loops?

Solution:

Questions:

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### Question 2.1

The piano strings that vibrate with the lowest frequencies consist of a steel wire around which a thick coil of copper wire is wrapped. Only the inner steel wire is under tension. What is the purpose of the copper coil?

Questions:

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### Problem 2.4

A piano string of length 1.50 m and mass density 25.0 mg/m vibrates at a (fundamental) frequency of 450.0 Hz.

1. What is the speed of the transverse string waves?
2. What is the tension?
3. What are the wavelength and frequency of the sound wave in air produced by vibration of the string? The speed of sound in air at room temperature is 340 m/s.

Solution:

Questions:

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### Problem 2.5

One of the harmonics of a column of air open at both ends has a frequency of 324 Hz and the next higher harmonic has a frequency of 378 Hz.

1. What the frequency of the next higher harmonic?
2. What is number, n, of this harmonic?
3. What is the fundamental frequency of the air column?

Solution:

• First,find the fundamental frequency $f_{1}$:
• $f_{n+1}-f_{n}=f_{1}=$ $378\,{\text{Hz}}-324\,{\text{Hz}}$ $=54\,{\text{Hz}}$
• The next highest harmonic frequency is found by adding the fundamental frequency:
• $f_{n+2}=f_{n+1}+f_{1}=378\,{\text{Hz}}+54\,{\text{Hz}}=432\,{\text{Hz}}$
• The next highest harmonic has a frequency of 432 Hz.
• You can find out which harmonic this is by dividing:
• $n=\frac{f_{n+2}}{f_{1}}=\frac{432\,{\text{Hz}}}{54\,{\text{Hz}}}=8$
• This is the eighth harmonic.

Questions:

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### Questions 2.2-2.3

• If the length of a guitar string is decreased while the tension remains constant, what happens to each of these quantities?
1. the wavelength of the fundamental
2. the frequency of the fundamental
3. the time for a pulse to travel the length of the string
4. the maximum velocity for a point on the string (assuming the amplitude is the same both times)
5. the maximum acceleration for a point on the string (assuming the amplitude is the same both times)
• A cello player can change the frequency of the sound produced by her instrument by
1. increasing the tension in the string,
2. pressing her finger on the string at different places along the fingerboard, or
3. bowing a different string.
• Explain how each of these methods affects the frequency.

Questions:

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### Problem 2.6

A wave on a string has equation $y(x,t) = (4.0\,{\text{mm}}) \sin (\omega t – kx)$, where $\omega = 6.0 \times 10^2\,$rad/s and $k = 6.0\,$ rad/m.

1. What is the amplitude of the wave?
• The amplitude of the wave is 0.004 m.
2. What is the wavelength?
• The wavelength is given by:
• $\lambda=\frac{2\pi}{k}=\frac{2\pi\,{\text{rad}}}{6\,\frac{\text{rad}}{\text m}}=1.0$5 m
3. What is the period?
• $T=\frac{1}{f}=\frac{2\pi}{\omega}=\frac{2\pi\,{\text{rad}}}{600\,\frac{\text{rad}}{\text m}}=1.05\times 10^{-2}$ s
4. What is the wave speed?
• $\nu=\lambda f=$ $\left(1.05\,{\text m}\right)\left(\frac{1}{1.05\times 10^{-2}\,{\text s}}\right)=$ 100 $\frac{\text m}{\text s}$
5. In which direction does the wave travel?
• The wave travels in the positive $x$ direction. We know this because $kx$ and $-\omega t$'s signs are not the same.

Questions:

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### Problem 2.7

1. Plot a graph for $y(x,t) = 4.0\,{\text{cm}}\cdot\sin\!\big(378\,{\text s}^{-1}\!\cdot\!t$ $–\, 314\,{\text m}^{-1}\!\cdot\!x\big)$, where $t$ is in s and $x$ and $y$ are in cm, versus $x$ at $t\!=\!0$ and at $t\!=\!\frac{1}{480}\,$s and find the wavelength of the wave.
2. For the same function, plot a graph of $y(x,t)$ versus $t$ at $x\!=\!0$ and find the period of the vibration.
3. Calculate the speed of the wave.

Solution:

Questions:

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### Problem 2.8

During a concert a single singer generates an intensity level of 55 dB at a certain location in the concert hall. With the whole choir singing the intensity level is 75 dB. Assuming that each singer generates the same intensity level, how many people are in the choir?

Solution:

Relevant Information/equations

1. $\beta_{\text{single}}=55$ dB
2. $\beta_{\text{choir}}=75$ dB
3. $\beta=10\log(\frac{I}{I_{0}})$
4. $I_{0}=10^{-12}\,\frac{\text W}{{\text m}^2}$

From 1, 3, and 4:

• $55\,{\text{dB}}=10\log(\frac{I}{I_{0}})\Leftrightarrow 10^{5.5}=\frac{I}{I_{0}}\Leftrightarrow\,I=10^{17.5}$

From 2, 3, 4 and the previous result:

• $75\,{\text{dB}}=10\log(\frac{I'}{I_{0}})$ where $I'=Ix$ (and $x$ is the unknown number of choir members)

$\Rightarrow10^{7.5}=x\frac{(10^{5.5})}{10^{-12}}\Leftrightarrow\,x=\frac{10^{7.5}}{10^{5.5}}=100$ choir members.

Questions:

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### Questions 2.4-2.8

• Why must astronauts on the surface of the moon communicate with each other by radio? Can they hear another spacecraft as it lands nearby? Can they hear by touching helmets?
• Astronauts on the surface of the moon communicate by radio because their voices cannot travel through the vacuum of space. Sound waves are mechanical waves, and as such they require a physical medium through which to propagate; radio waves are electromagnetic waves, and can propagate through a vacuum… but more about that later. Applying this logic, our astronauts would not be able to hear a spacecraft landing nearby. They could conceivably communicate by touching helmets, as the helmets-in-contact constitute a physical medium.
• When an earthquake occurs, the S waves (transverse waves) are not detected on the opposite side of the Earth while the P waves (longitudinal waves) are. How does this provide evidence that the Earth’s solid core is surrounded by liquid?
• Why is it that your own voice sounds strange to you when you hear it played back on a tape recorder, but your friends all agree that it is just what your voice sounds like? [Hint: Consider the media through which the sounds wave travels when you usually hear your own voice.]
• Is the vibration of a string in a piano, guitar, or violin a sound wave? Explain.
• The vibration of a string on a stringed instrument is not a sound wave, but a standing wave on a string. A sound wave is created when the air surrounding the string is pushed back and forth by the string with the same frequency as the string's vibration.
• Many real estate agents have an ultrasonic rangefinder that enables them to quickly and easily measure the dimensions of a room. The device is held to one wall and reads the distance to the opposite wall. How does it work?
• The rangefinder works by producing a high-frequency sound wave, which reflects off of the opposite wall and travels back to the rangefinder in time $t=2L/c$, where $L$ is the length of the room and $c$ is the speed of sound in air. If the time it takes the sound wave to return to the source can be measured accurately, $L$ can easily be solved for.

Questions:

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# Workshop 1 (Apr 2,3)

## Oscillations

### Equations and Relations WS.1

1. Frequency, period, angular frequency
• $f=\frac{1}{T}$
• $\omega=2\pi f$
2. Position, velocity and acceleration of a harmonic oscillator
• $x=A\cos\left(\frac{2\pi\cdot t}{T}\right)=A\cos(\omega t)$
• $v=-A\omega\sin(\omega t)$
• $v_\text{max}=A\omega$
• $a=-A\omega^2\cos(\omega t)$
• $a_\text{max}=A\omega^2$
3. Spring period
• $T=2\pi\sqrt{\frac{m}{k}}$
4. Energy
• $E=\frac{1}{2}kA^2$
• $U=\frac{1}{2}kA^2\cos^2(\omega t)$
• $K=\frac{1}{2}kA^2\sin^2(\omega t)$
5. Simple pendulum period
• $T=2\pi\sqrt{\frac{L}{g}}$
6. Physical pendulum period
• $T=2\pi\sqrt{\frac{I}{mgd}}$
7. Underdamping
• $A=A_0e^{-\frac{bt}{2m}}$
8. Hooke's law:
• $F=-kx$

### Problems and Questions

##### Problem 1.1

A mass of 1 kg is attached to a spring and undergoes simple harmonic oscillations with a period of 1 s. What is the force constant of the spring?

Solution:

Questions:

• $k=39.48\,\frac{\text N}{\text m}$, since $k=\frac{m}{\big(T/(2\pi)\big)^2}$ $=\frac{1.0\,{\text{kg}}}{\big(1.0\,{\text s}/(2\pi)\big)^2}=39.48$ N/m (now the units are correct)

##### Problem 1.2
1. If a simple pendulum has period T = 1.0 s and you double its length, what is its new period in terms of T ?
2. If a simple pendulum has a length L = 1.0 m and you want to triple its frequency, what should be its length?
3. Suppose a simple pendulum has a length L and period T on earth. If you take it to a planet where the acceleration of freely falling objects is ten times what it is on earth, what should you do to the length to keep the period the same as on earth?
4. If you do not change the simple pendulum’s length in the previous part, what is its period on that planet in terms of T ?
5. If a simple pendulum has a period T and you triple the mass of its bob, what happens to the period (in terms of T)?

Solution:

Questions:

• The period increase by a factor of $\sqrt{2}$

##### Problem 1.3

What is the period of a pendulum formed by placing a horizontal axis (i.e. pivot point)

1. through the end of a meterstick (100-cm mark)?
2. through the 75-cm mark?
3. through the 60-cm mark?

Assume g = 9.80 m/s2.
Hint: $I_\text{cm}=\frac{1}{12}ml^2$ , Parallel axis theorem: $I=I_\text{cm}+md^2$

Solution:

Questions:

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##### Questions 1.1-1.3
• A pendulum clock runs too slow and loses time. What adjustment should be made?
• How can the principle of the pendulum be used to compute (a) length, (b) mass, and (c) time?
• A pendulum is mounted in an elevator that moves upward with constant acceleration. Is the period greater than, less than, or the same as when the elevator is at rest? Why?

Questions:

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##### Problem 1.4

A 1.5-kg mass oscillates at the end of a spring in SHM. The amplitude of the vibration is 0.15 m, and the spring constant is 80 N/m. If the mass is displaced 15 cm,

• what are the magnitude and direction of the acceleration and force on the mass?

If the system is now operated on a frictionless horizontal surface,

• what is total energy?
• what is the maximum velocity?
• what is the maximum acceleration?

Solution:

Questions:

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##### Questions 1.4-1.7
• What effect will doubling the amplitude A of a body moving with SHM have on (a) the period, (b) the maximum velocity, and (c) the maximum acceleration?
• Explain how the period of a mass-spring system can be independent of amplitude, even though the distance traveled during each cycle is proportional to the amplitude.
• Explain why the velocity in SHM is greatest when the magnitude of the acceleration is the least.
• A mass hanging vertically from a spring and a simple pendulum both have a period of oscillation of 1 s on Earth. An astronaut takes the two devices to another planet where the gravitational field is stronger than that of Earth. For each of the two systems, state whether the period is now longer than 1 s, shorter than 1 s, or equal to 1 s. Explain your reasoning.

Questions:

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##### Problem 1.5

5. A mass is vibrating at the end of a spring of force constant 225 N/m. The figure shows a graph of its position x as a function of time t.

1. At what times is the mass not moving?
2. How much energy did the system originally contain?
3. How much energy did the system lose between t = 1.0 s and t = 4.0 s

Solution:

Questions:

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