Assign 24

course Phy 232

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024. `Query 13

Physics II

05-05-2007

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11:25:04

query experiment to be viewed and read but not performed: transverse and longitudinal waves in aluminum rod

what is the evidence that the higher-pitched waves are longitudinal while the lower-pitched waves are transverse?

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RESPONSE -->

lower pitched wave was dampened by touching the side where the higher pitched wave was dampened by touching the end

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11:25:30

STUDENT RESPONSE: The logitudinal waves had a higher velocity.

That doesn't provide evidence that the high-pitched wave was longitudinal, since we didn't directly measure the velocity of those waves. The higher-pitches waves were damped out much more rapidly by touching the very end of the rod, along its central axis, than by touching the rod at the end but on the side.

The frequency with which pulses arrive at the ear determines the pitch.

The amplitude of the wave affects its intensity, or energy per unit area. For a given pitch the energy falling per unit area is proportional to the square of the amplitude.

Intensity is also proportional to the square of the frequency. **

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good

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11:25:36

Query General College Physics and Principles of Physics 12.08: Compare the intensity of sound at 120 dB with that of a whisper at 20 dB.

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11:25:47

The intensity at 120 dB is found by solving the equation dB = 10 log(I / I_threshold) for I.

We get

log(I / I_threshold) = dB / 10, so that

I / I_threshold = 10^(120 / 10) = 12and

I = I_threshold * 10^12.

Since I_threshold = 10^-12 watts / m^2, we have for dB = 120:

I = 10^-12 watts / m^2 * 10^12 = 1 watt / m^2.

The same process tells us that for dB = 20 watts, I = I_threshold * 10^(20 / 10) = 10^-12 watts / m^2 * 10^2 = 10^-10 watts / m^2.

Dividing 1 watt / m^2 by 10^-10 watts / m^2, we find that the 120 dB sound is 10^10 times as intense, or 10 billion times as intense.

A more elegant solution uses the fact that dB_1 - dB_2 = 10 log(I_1 / I_threshold) - ( 10 log(I_2 / I_threshold) )

= 10 log(I_1 / I_threshold) - ( 10 log(I_2 / I_threshold) )

= 10 {log(I_1) - log( I_threshold) - [ ( log(I_2) - log(I_threshold) ]}

= 10 { log(I_1) - log(I_2)}

= 10 log(I_1 / I_2).

So we have

120 - 20 = 100 = 10 log(I_1 / I_2) and

log(I_1 / I_2) = 100 / 10 = 10 so that

I_1 / I_2 = 10^10.

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11:25:55

Query gen phy 12.30 length of open pipe, 262 Hz at 21 C? **** gen phy What is the length of the pipe?

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11:26:18

GOOD STUDENT SOLUTION

First we must determine the velocity of the sound waves given the air temperature. We do this using this formula

v = (331 + 0.60 * Temp.) m/s

So v = (331 + 0.60 * 21) m/s

v = 343.6 m/s

The wavelength of the sound is

wavelength = v / f = 343.6 m/s / (262 Hz) = 0.33 meters.

So 262 Hz = 343.6 m/s / 4 * Length

Length = 0.33 meters

f = v / (wavelength)

262 Hz = [343 m/s] / (wavelength)

wavelength = 1.3 m.

So the wavelength is 1.3 m. If it's an open pipe then there are antinodes at the ends and the wavelength is 2 times the length, so length of the the pipe is about 1.3 m / 2 = .64 m, approx..

Had the pipe been closed at one end then there would be a node and one end and an antinode at the other and the wavelength of the fundamental would have therefore been 4 times the length; the length of the pipe would then have been 1.3 m / 4 = .32 m. **

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11:28:38

**** Univ phy 16.72 (10th edition 21.32):  Crab nebula 1054 A.D.;, H gas, 4.568 * 10^14 Hz in lab, 4.586 from Crab streamers coming toward Earth.  Velocity?  Assuming const vel diameter?    Ang diameter 5 arc minutes; how far is it?

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fr = fs * sqrt((c-v)/(c+v))

(c-v)/(c+v) = (fr/fx)^2

(3 x 10^8 - v)/(3 x 10^8 + v) = (4.586 x 10^14/4.568 x 10^14)^2

(4.586 x 10^14/4.568 x 10^14)^2= 1.0079

3.0237 x 10^8 + 1.0079v = 3 x 10^8 - v

2.0079v = 2.37 x 10^6

v = 1.18 x 10^6

949 * 365 * 24 * 60 * 60 = 2.99 x 10^10 * 1.18 x 10^6 = 3.53 x 10^16 * 2 = 7.06 x 10^16 m = 7.46 LY (using online converter)

5/60 = 1/12

circumference is pi (7.06 x 10^16) and 1/12 of that is 5.88 x 10^9

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11:29:44

** Since fR = fS ( 1 - v/c) we have v = (fR / fS - 1) * c = 3 * 10^8 m/s * (4.586 * 10^14 Hz) / (4.568 * 10^14 Hz) = 1.182 * 10^6 m/s, approx.

In the 949 years since the explosion the radius of the nebula would therefore be about 949 years * 365 days / year * 24 hours / day * 3600 seconds / hour * 1.182 * 10^6 m/s = 3.5 * 10^16 meters, the diameter about 7 * 10^16 meters.

5 minutes of arc is 5/60 degrees or 5/60 * pi/180 radians = 1.4 * 10^-3 radians. The diameter is equal to the product of the distance and this angle so the distance is

distance = diameter / angle = 7 * 10^16 m / (1.4 * 10^-3) = 2.4 * 10^19 m.

Dividing by the distance light travels in a year we get the distance in light years, about 6500 light years.

CHECK AGAINST INSTRUCTOR SOLUTION: ** There are about 10^5 seconds in a day, about 3 * 10^7 seconds in a year and about 3 * 10^10 seconds in 1000 years. It's been about 1000 years. So those streamers have had time to move about 1.177 * 10^6 m/s * 3 * 10^10 sec = 3 * 10^16 meters.

That would be the distance of the closest streamers from the center of the nebula. The other side of the nebula would be an equal distance on the other side of the center. So the diameter would be about 6 * 10^16 meters.

A light year is about 300,000 km/sec * 3 * 10^7 sec/year = 9 * 10^12 km = 9 * 10^15 meters. So the nebula is about 3 * 10^16 meters / (9 * 10^15 m / light yr) = 3 light years in diameter, approx.

5 seconds of arc is 5/60 of a degree or 5 / (60 * 360) = 1 / 4300 of the circumference of a full circle, approx.

If 1/4300 of the circumference is 6 * 10^16 meters then the circumference is about 4300 times this distance or about 2.6 * 10^20 meters.

The circumference is 1 / (2 pi) times the radius. We're at the center of this circle since it is from here than the angular diameter is observed, so the distance is about 1 / (2 pi) * 2.6 * 10^20 meters = 4 * 10^19 meters.

This is about 4 * 10^19 meters / (9 * 10^15 meters / light year) = 4400 light years distant.

Check my arithmetic. **

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got the velocity right, but where'd that equation come from, it would have been much easier than the one in the book

The equation used in the given solution applies when v is much less than c. It can be derived using the following facts:

The square root of a number very close to 1 is very nearly twice as close to 1.

When you divide by a number close to 1 the result is equally close to 1. If the divisor is less than 1 the result is greater than the dividend, and vice versa.

(c-v) / (c+v) = (1 - v/c) / (1 + v/c), where v/c is small.

1 + v/c is bigger than 1 and 1 - v/c is close to 1; when (1 - v/c) is divided by (1 + v/c) the result is about v/c less than 1 - v/c, so is about 1 - 2 v/c.

The square root of 1 - 2 v/c is twice as close to 1, or about 1 - v/c.

So as long as v < < c

fr = fs * sqrt((c-v)/(c+v)) is close to

fr = fs * (1 - v/c).

For example if v/c is .01 then 1 + v/c is 1.01 and 1 - v/c is .99.

If you divide .99 by 1.01 the result is less than .99 be about .01 * .99 = .0099, giving result about .9801, very close to .98.

So (v - c) / (v + c) = .98.

sqrt(.98) = .99.

diameter I got

also, didn't get the trig part right.....need to remediate some of that

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11:31:29

**** query univ phy 16.66 (21.26 10th edition). 200 mHz refl from fetal heart wall moving toward sound; refl sound mixed with transmitted sound, 85 beats / sec. Speed of sound 1500 m/s.

What is the speed of the fetal heart at the instant the measurement is made?

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beat is 2 waves of close freq superposed on each other.

one beat is 200 MHz, 85 beats is the diff. between two feqs so superposed wave has freq 200,000,085

(equation 16.26) fl = fs * (1 + (vL/v))

fl = 2 MHz * (1 + (vl/1500))

reflected sound is fs = (1 + vs/v) * 2 MHz = fl of prev. equation

so fl = [(1 + vs/1500) * 2 x 10^8] * (1 + (vs/v))

200000085 = [(1 + vs/1500) * 2 x 10^8] * (1 + vs/1500)

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11:32:15

. ** 200 MHz is 200 * 10^6 Hz = 2 * 10^8 Hz or 200,000,000 Hz.

The frequency of the wave reflected from the heart will be greater, according to the Doppler shift.

The number of beats is equal to the difference in the frequencies of the two sounds. So the frequency of the reflected sound is 200,000,085 Hz.

The frequency of the sound as experienced by the heart (which is in effect a moving 'listener') is fL = (1 + vL / v) * fs = (1 + vHeart / v) * 2.00 MHz, where v is 1500 m/s.

This sound is then 'bounced back', with the heart now in the role of the source emitting sounds at frequency fs = (1 + vHeart / v) * 2.00 MHz, the 'old' fL. The 'new' fL is

fL = v / (v - vs) * fs = v / (v - vHeart) * (1 + vHeart / v) * 2.00 MHz.

This fL is the 200,000,085 Hz frequency. So we have

200,000,085 Hz = 1500 m/s / (v - vHeart) * (1 + vHeart / v) * 2.00 MHz and

v / (v - vHeart) * (1 + vHeart / v) = 200,000,085 Hz / (200,000,000 Hz) = 1.000000475.

A slight rearrangement gives us

(v + vHeart) / (v - vHeart) = 1.000000475 so that

v + vHeart = 1.000000475 v - 1.000000475 vHeart and

2.000000475 vHeart = .000000475 v, with solution

vHeart = .000000475 v / (2.000000475), very close to

vHeart = .000000475 v / 2 = .000000475 * 1500 m/s / 2 = .00032 m/s,

about .3 millimeters / sec. **

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just the logic of this one makes my brain hurt!!

I've seen a ton of untrasounds and they look really cool!

I've only seen the two or three they've done on me over the last 20 years or so. Amazing stuff. Resolution hasn't changed much over that time--still using the same frequencies (wavelengths around 1/3 mm, if I recall correctly; if I'm right that would correspond to about a megaHertz).

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Good work. See my notes and let me know if you have additional questions.