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exam_1_review [2014/04/19 07:18]
wikimanager [Review problem 3] added solutions
exam_1_review [2014/04/19 19:26] (current)
wikimanager [Review problem 1]
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   * a) Which of these waves travel in the $+x$ direction? (3 pts)   * a) Which of these waves travel in the $+x$ direction? (3 pts)
 +    * <color green>​waves 1 and 4 </​color>​
 +      * <color green>​(look for opposite signs in front of the //x// and //t// terms inside the cos or sin function)</​color>​
   * b) Which of these waves have the same wavelength as wave 1?  (3 pts)   * b) Which of these waves have the same wavelength as wave 1?  (3 pts)
 +    * <color green>​wave 4 </​color>​
 +      * <color green>​(look for the same magnitude of the coefficient in front of //x// inside the cos or sin function)</​color>​
   * c) Which of these waves have the same amplitude as wave 2? (3 pts)   * c) Which of these waves have the same amplitude as wave 2? (3 pts)
 +    * <color green>​wave 5 </​color>​
 +      * <color green>​(look for the same magnitude of the coefficient in front of the cos or sin function)</​color>​
   * d) Which of these waves have the same period as wave 3? (3 pts)   * d) Which of these waves have the same period as wave 3? (3 pts)
 +    * <color green>​wave 1 </​color>​
 +      * <color green>​(look for the same magnitude of the coefficient in front of //t// inside the cos or sin function)</​color>​
   * e) Which of these waves have the same speed as wave 4? (3 pts)   * e) Which of these waves have the same speed as wave 4? (3 pts)
 +    * <color green>​wave 5 </​color>​
 +      * <color green>​(look for the same magnitude of the ratio of the coefficients in front of //t// and in front of //x// inside the cos or sin function)</​color>​
 +
 +<color blue>See [[start#​equation_sheet_ch14|Equation sheet for Ch. 14, bullet #6]] for details...</​color>​
 +
 +===All of the parameters for each wave are summarized below===
 +^  Wave  ^  Direction ​ ^  Amplitude (m) ^  Wavelength (m)  ^  Period (s)  ^  Frequency (Hz)  ^  Magnitude of speed $\left(\frac{\mathbf m}{\mathbf s}\right)$ ​ ^
 +|  1   ​| ​ $+x$     ​| ​  ​0.12 ​ |  $\frac{2\pi}{3}$ ​ |  $\frac{2\pi}{21}$ ​ |  $\frac{21}{2\pi}$ ​ |  7     |
 +|  2   ​| ​ $-x$     ​| ​  ​0.15 ​ |  $\frac{\pi}{3}$ ​  ​| ​ $\frac{\pi}{21}$ ​  ​| ​ $\frac{21}{\pi}$ ​  ​| ​ 7     |
 +|  3   ​| ​ $-x$     ​| ​  ​0.13 ​ |  $\frac{\pi}{3}$ ​  ​| ​ $\frac{2\pi}{21}$ ​ |  $\frac{21}{2\pi}$ ​ |  3.5   |
 +|  4   ​| ​ $+x$     ​| ​  ​0.27 ​ |  $\frac{2\pi}{3}$ ​ |  $\frac{\pi}{6}$ ​   |  $\frac{6}{\pi}$ ​   |  4     |
 +|  5   ​| ​ $-x$     ​| ​  ​0.15 ​ |  $\frac{2\pi}{9}$ ​ |  $\frac{\pi}{18}$ ​  ​| ​ $\frac{18}{\pi}$ ​  ​| ​ 4     |
  
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 {{ :​figs:​ex1rev2.jpg?​nolink&​189|}} A musician (Andrew Bird) used a double spinning horn speaker during a recent tour. While one horn spins toward you, the other spins away. Say the horns are emitting a frequency of 880 Hz, are spinning with an angular velocity of 2.0 rad/s, and that each horn is 1.0 m long.  {{ :​figs:​ex1rev2.jpg?​nolink&​189|}} A musician (Andrew Bird) used a double spinning horn speaker during a recent tour. While one horn spins toward you, the other spins away. Say the horns are emitting a frequency of 880 Hz, are spinning with an angular velocity of 2.0 rad/s, and that each horn is 1.0 m long. 
   * a) Due to a Doppler shift, what is the greatest pitch (frequency $f$) you will hear? Will this be when the horn is spinning towards you or away from you? (Remember, $v_t = A\omega$, where $A$ is the distance from the rotation axis to the opening of the horn) (3 pts)   * a) Due to a Doppler shift, what is the greatest pitch (frequency $f$) you will hear? Will this be when the horn is spinning towards you or away from you? (Remember, $v_t = A\omega$, where $A$ is the distance from the rotation axis to the opening of the horn) (3 pts)
 +    * <color green>​Greatest pitch - motion towards you</​color>​
 +    * <color green>​$f'​=f\left(\frac{1}{1-\frac{u}{v}}\right)$ $=f\left(\frac{1}{1-\frac{A\omega}{v}}\right)$ $=880\,​{\text{Hz}}\,​\left(\frac{1}{1-\frac{1.0\,​{\text m}\,​\cdot\,​2.0\,​\frac{\text{rad}}{\text s}}{340\,​\frac{\text m}{\text s}}}\right)$ $=885\,​$Hz ​ </​color>​
   * b) What is the lowest pitch you will hear from the speakers due to a Doppler shift? Will this be when the horn is spinning towards you or away from you? (3 pts)   * b) What is the lowest pitch you will hear from the speakers due to a Doppler shift? Will this be when the horn is spinning towards you or away from you? (3 pts)
 +    * <color green>​Lowest pitch - motion away from you</​color>​
 +    * <color green>​$f'​=f\left(\frac{1}{1+\frac{u}{v}}\right)$ $=f\left(\frac{1}{1+\frac{A\omega}{v}}\right)$ $=880\,​{\text{Hz}}\,​\left(\frac{1}{1+\frac{1.0\,​{\text m}\,​\cdot\,​2.0\,​\frac{\text{rad}}{\text s}}{340\,​\frac{\text m}{\text s}}}\right)$ $=875\,​$Hz ​ </​color>​
   * c) What is the beat frequency you hear from this instrument? (3 pts)   * c) What is the beat frequency you hear from this instrument? (3 pts)
 +    * <color green> $f_\text{beat}=\big|\,​f_1-f_2\big|$ $=\big|885\,​{\text{Hz}}-875\,​{\text{Hz}}\big|$ $=10\,​$Hz ​  </​color>​
   * d) At the concert your friend sitting 1 m from the speakers hears an intensity of $9.0\times 10^{-2}\frac{\text W}{\,{\text m}^2}$. What intensity do you hear sitting 10 m away from the speakers? (3 pts)   * d) At the concert your friend sitting 1 m from the speakers hears an intensity of $9.0\times 10^{-2}\frac{\text W}{\,{\text m}^2}$. What intensity do you hear sitting 10 m away from the speakers? (3 pts)
 +    * <color green> $I=\frac{\text{Power}}{4\pi r^2}$   </​color>​
 +    * <color green> $\frac{I_\text{friend}}{I_\text{you}}=\frac{(10\,​{\text m})^2}{(1\,​{\text m})^2}$ $=100$ ​   </​color>​
 +    * <color green> $I_\text{you}=\frac{1}{100}I_\text{friend}$ $=9.0\times 10^{-4}\frac{\text W}{\,{\text m}^2}$ ​  </​color>​
   * e) What is the intensity level in decibels that your friend hears? (3 pts)   * e) What is the intensity level in decibels that your friend hears? (3 pts)
 +    * <color green> $\beta=10\,​{\text{dB}}\log\left(\frac{I_1}{I_\text{t.h.}}\right)$ $=10\,​{\text{dB}}\log\left(\!\frac{9.0\times 10^{-2}\frac{\text W}{\,{\text m}^2}}{ 10^{-12}\frac{\text W}{\,{\text m}^2}}\!\right)$ $\approx 110\,​$dB ​ </​color>​
  
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 The air pressure variations in a sound wave cause the eardrum (tympanic membrane) ​ to vibrate. The air pressure variations in a sound wave cause the eardrum (tympanic membrane) ​ to vibrate.
   - For a given vibration amplitude, are the maximum velocity and acceleration of the eardrum greatest for high frequency sounds or low frequency sounds? (1 pts)   - For a given vibration amplitude, are the maximum velocity and acceleration of the eardrum greatest for high frequency sounds or low frequency sounds? (1 pts)
 +    * <color green>​$v_\text{max}$ $=x_\text{max}\omega$</​color>​
 +    * <color green>​$a_\text{max}$ $=x_\text{max}\omega^2$</​color>​
 +    * <color green>​the greatest values are for the high-frequency sound</​color>​
   - Find the maximum velocity and the maximum acceleration of the eardrum for vibrations of amplitude $1.0\times 10^{-8}\,$m at a frequency of 20.0 kHz. (5 pts)    - Find the maximum velocity and the maximum acceleration of the eardrum for vibrations of amplitude $1.0\times 10^{-8}\,$m at a frequency of 20.0 kHz. (5 pts) 
 +    * <color green>​$v_\text{max}$ $=x_\text{max}\omega$ $=x_\text{max}(2\pi f)$ $=1.0\times 10^{-8}\,​$m$\,​\cdot\,​6.283\cdot 20\times 10^3\,$Hz $=1.26\times 10^{-3}\frac{\text m}{\text s}$ </​color>​
 +    * <color green>​$a_\text{max}$ $=x_\text{max}\omega^2$ $=x_\text{max}(2\pi f)^2$ $=1.0\times 10^{-8}\,​$m$\,​\cdot\,​(6.283\cdot 20\times 10^3\,​$Hz$)^2=158\,​\frac{\text m}{\,{\text s}^2}$ </​color>​
   - What is the period of a complete oscillation of the ear drum at this frequency? (2 pts)   - What is the period of a complete oscillation of the ear drum at this frequency? (2 pts)
 +    * <color green>​$T=\frac{1}{f}$ $=\frac{1}{20\times 10^3\,​{\text{Hz}}}$ $=5.0\times 10^{-5}\,​$s</​color>​
   - Using a crude model of the eardrum as a mass (3.0 mg) on a spring, what would be the spring constant of the eardrum, assuming the resonance frequency of 20.0 kHz? (3 pts)   - Using a crude model of the eardrum as a mass (3.0 mg) on a spring, what would be the spring constant of the eardrum, assuming the resonance frequency of 20.0 kHz? (3 pts)
 +    * <color green>​$T=2\pi\sqrt{\frac{m}{k}}$</​color>​
 +    * <color green>​$\left(\frac{T}{2\pi}\right)^2=\frac{m}{k}$</​color>​
 +    * <color green>​$k=\frac{4\pi^2m}{T^2}$ $=\frac{4\,​\cdot\,​3.14159^2\,​\cdot\,​3.0\times 10^{-6}\,​{\text{kg}}}{\big(5.0\times 10^{-5}\,​{\text s}\big)^2}$ $=4.74\times 10^4\frac{\text N}{\text m}$</​color>​
   - The ear canal (external auditory canal) can be modeled as a tube with one closed end. If the length of the ear canal is 25 mm long and the speed of sound in air is 340 m/s, what is the fundamental (1<​sup>​st</​sup>​ harmonic) of the ear canal? (4 pts)    - The ear canal (external auditory canal) can be modeled as a tube with one closed end. If the length of the ear canal is 25 mm long and the speed of sound in air is 340 m/s, what is the fundamental (1<​sup>​st</​sup>​ harmonic) of the ear canal? (4 pts) 
 +    * <color green>​$f_1=\frac{v}{4L}$ $=\frac{340\,​\frac{\text m}{\text s}}{4\,​\cdot\,​25\times 10^{-3}\,​{\text m}}$ $=3400\,​$Hz</​color>​
  
exam_1_review.txt ยท Last modified: 2014/04/19 19:26 by wikimanager