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course PHY 232
10:53 am 3/18
110307 Physics`q001. The log of a number is the power to which you need to raise 10 to get that number. What are the logs of the following numbers?
• 100
• 10000
• .0001
• 10 000 000 000 000
• 10^5 / 10^14
• 100 / 10^7
****2
4
-4
13
-9
-6
@& @& That last one would be 10^-5, with log -5*@
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`q002. Quickly sketch the following graphs
• log(x) vs. x from x = 1 to x = 100
• log(x) vs. x from x = 10 to x = 1 000
• log(x) vs. x from x = 100 000 to 10 000 000
• log(x) from x = 0.1 to x = 10.
What do your graphs have in common?
****They all are concave down and increasing at a decreasing rate. They have the same proportions as well.
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`q003. Using your graphs from before, estimate the following:
• log(3)
• log(5)
• log(300)
• log(500)
• log(2 000 000)
****0.4
0.7
2.5
2.7
6.3
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`q004. Relabel one or more of your graphs to estimate each of the following:
• log(300 000 000)
• log(5 000 000 000)
• log(.00003)
**** 8.5
8.69
-4.5
This last one was a trick in that the y axis was relabelled as negatives, that is -4 was at 2 and -5 at 1 so y went down .
@& .0003 is between 10^-4 and 10^-3, so the log is between -4 and -3.*@
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`q005. The hearing threshold is defined to be 10^-12 watts / m^2. We can stand sounds up to intensity 1 watt / m^2 without pain. 1 watt / m^2 is called the pain threshold.
How many times more intense than the hearing threshold is the pain threshold?
What is the log of this ratio?
****1/10^-12 = 10^12
log = 12
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`q006. Based on your graph(s), what is the log of the ratio of each of the following intensities to the hearing threshold?
• .0001 watts / m^2
• 10^-8 watts / m^2
• .03 watts / m^2
• 5 * 10^-7 watts / m^2
****8
4
10.5
5.7
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`q007. If the log of the ratio of intensity to hearing threshold intensity is the given number, then what is the intensity?
• 5
• 9
• 3.5
****10^-7
10^-3
5*10^-13
@& @& The last would be between 10^-8 and 10^-9, closer to 10^-9.*@
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`q008. The decibel level of a sound is 10 times the log of the ratio of its intensity to the hearing threshold intensity. What is the decibel level of each of the following, which you have seen in a previous question:
• .0001 watts / m^2
• 10^-8 watts / m^2
• .03 watts / m^2
• 5 * 10^-7 watts / m^2
****80
40
105
57
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`q009. What is the intensity of a sound with each of the following decibel levels?
• 80 dB
• 25 dB
• 53 dB
• 110 dB
****
0.0001
3.2*10^-10
2*10^-7
0.1
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`q010. What is the ratio of the intensities of two sounds whose decibel levels differ by 33 dB?
****
A simple derivation:
`d B= 33 dB = 10 dB log (I_2/I-0) - 10 dB log (I_1/I-0)
3.3 = log ((I_2/I-0)/(I_1/I-0)) = log I_2/I_1
Thus
I_2/I_1= 10^3.3 ≈ 1995.3
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University Physics:
`q011. If the waveforms y_1 = A cos(omega * t) and y_2 = A cos( (omega + `dOmega) t are mixed, what is the equation of the combined wave function, in terms of sines and cosines of omega * t and `dOmega * t?
What is the maximum amplitude of the resulting beats?
****
y = A cos(omega*t) + A cos(omega+ `d omega)*t = A cos(omega*t) + A cos(omega*t)*cos( `d omega*t) + A sin(omega*t)*sin( `d omega*t)
y = A cos omega*t ( 1+ cos `d omega*t) + Asin(omega*t)*sin( `d omega*t)
In class it seems we reduced this further but my notes and the notes online are incomplete and the y_1/y_2 were sin not cos. Another route, and the one the book uses, is:
cos a + cos b = 2 cos(a/2-b/2)cos(a/2+b/2)
then
y= 2A*cos( `d omega*t/2)*cos(omega*t+ `d omega*t/2)
Which could also be:
y = 2A(cos(omega*t+ `d omega*t) + cos omega*t) and this is just like above but with 2A.
Max Amp = 2A
@& Good, but a simpler answer is possible. Beats occur when the 'peaks' of the waves match up. Since they both have amplitude A, when the peaks match the resulting amplitude will be 2 A.*@
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`q012. A speaker suspended by its power cord oscillates back and forth with an amplitude of 10 cm and a frequency of 40 cycles / minute. It emits sound at 1000 Hz. A microphone is mounted in front of the speaker, and another behind it, so that when the speaker moving toward one microphone it is moving away from the other. Both speakers are at some distance from the speaker.
The sound collected by the microphones is mixed into a single sound.
What is the maximum frequency of the resulting beats?
****
Max f occurs when there is the largest difference between the two frequencies detected by the mics. The speaker has SHM where x = 0.1 cos (80π*t) and v = -8π sin (80π*t) which has max of v_max = 8π m/s and it occurs when the speaker is at eq point, or mid swing. So
f_mic1 = 340/(340+8π)*f_s= 931.2 Hz
and
f_mic_2 = 340/(340-8π)*f_sp= 1079.8 Hz
f_beats = 1079.8 - 931.2 = 148.6 Hz
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@& RIght approach, but I think you have the speaker velocity wrong.
For the motion of the speaker omega = 4 / 3 pi rad/sec, so vMax = omega * A = .1 m * 4/3 pi rad/sec = .4 m/s, very roughly.
That results in about a .12% change in frequency, each way, so the mixed sounds will differ in frequency by about .24%.
.24% of 1000 Hz is about 2.4 Hz.*@
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&#Your work looks good. See my notes. Let me know if you have any questions. &#