A wire is made out of a special steel alloy with Young\'s modulus Y=2.0*10^11, d

Question

A wire is made out of a special steel alloy with Young's modulus Y=2.0*10^11, density =7800 kg/m^3, radius 1.00 mm, and coefficient of linear thermal expansion =2.50×10^5 K^1. The initial temperature of the wire is 20°C. As a constant tension of 3.53 x 10^4 N is applied to the wire, the wire is measured to be exactly 60.0 cm long. (a) What is the fundamental frequency of the wire? Assume the speed of sound is 344 m/s. (b) Where should you pluck the wire to achieve the third harmonic (second overtone) without also exciting the fundamental mode, and what is its frequency? (c) If the wire is heated to the melting point of steel (1363 oC), by how much does the fundamental frequency change?

Explanation / Answer

According to the question provided we need to find the following parts .

Part a) f=1/2L[T/d]^1/2

Plug in the values we get

1/2*0.6 [3.53*10^4/7800]^1/2 = 1.8 Hz ==============ANSWER)

Part b) we know that

third harmonic frequency=3f

=3*1.77429 =5.32287 Hz =======================ANSWER)

Part c) T=YA[ALPHA][t2-t1]

Plug in the values we get

2*10^11*3.14*1*10^-6*2.5*10^-5*[1363-20] = 21085.1 N

Now we can solve for the frequency

f=1/2*0.6 [21085.1/7800]^1/2

=1.37 Hz

fundamental frequency=1.8-1.37 =0.43 hz ========================ANSWER) PART C)

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