A 1 m long string has n mass per unit length of 2.0 times 10^-3 kg/m and is unde

Question

A 1 m long string has n mass per unit length of 2.0 times 10^-3 kg/m and is under a tension of 80 N. Find the first four harmonies of this string. Write the expressions for the frequencies of the first three harmonics for a pipe that is open at both ends, and for a pipe that is open at one end and closed at the other end. A steel wire in a piano has a length of 0.7 m and a mass of 4.3 times 10^-3 kg. To what tension must this wire be stretched in order that the fundamental vibration corresponds to middle C (f_c = 261.6 Hz on t lie chromatic musical scale)? A stone is dropped into a well. The splash is heard 3 seconds later. What is the depth of the well? A trunk is being driven towards a stationary car. If the truck emits sounds with frequency of 1000 Hz, and an observer in the stationary car detects a frequency of 1200 Hz, how fast is the truck traveling? A cop car drives at 30 m/s toward the scene of a crime, with its siren blaring at a frequency of 2000 Hz. At what frequency do people hear the siren as it approaches? At what frequency do they hear it as it passes? Extra Credit: What is Mach number? If a vehicle is labeled Mach 12, what does that mean? What does "breaking the sound barrier" mean? Who was the first person to break the sound barrier?

Explanation / Answer

1)

Frequency ,f = sqft(T/u)/(2L)

So, f = sqft(80/(0.002))/(2*1)

= 1.86*10^3 Hz <------ fundamental frequency

Second harmonic = 2*1.86*10^3 = 3.72*10^3 Hz

Third harmonic = 3*1.86*10^3 = 5.58*10^3 Hz

Fourth harmonic = 4*1.86*10^3 = 7.44*10^3 Hz

2)

For pipe open at both ends ,

fundamental frequency, f = v/(2L)

2nd harmonic = 2v/2L

3rd harmonic = 3v/2L

4th harmonic = 4v/2L

For closed organ pipe:

Fundamental harmonic = v/4L

2nd harmonic = 3v/4L

3rd harmonic = 5v/4L

3)

f = sqft(T/u)/2L

So, 261.6 = sqft(T/0.0043)/(20.7)

So, T = 126091 N

4)

Let depth of well be D

So, time taken to reach the bottom:

t1 = sqrt(2D/9.8)

Time taken for the splash to reach the top,

t2 = D/340

So, total time = t1 + t2 = 3

= sqrt(2D/9.8) + D/340 = 3

So, D = 40.7 m

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