A uniform wire of length 0.60\text{ m} and linear mass density 4.0 \times 10^{-3}\text{ kg… — JAMB UTME

A uniform wire of length $0.60\text{ m}$ and linear mass density $4.0 \times 10^{-3}\text{ kg/m}$ is fixed at both ends under a tension of $360\text{ N}$ . When plucked, the wire vibrates in its second overtone. This frequency is found to be in resonance with the first overtone of an air column in a pipe closed at one end. Taking the speed of sound in air as $340\text{ m/s}$ , what is the length of the pipe?

A
$0.17\text{ m}$
$0.34\text{ m}$ Cevap
C
$0.51\text{ m}$
D
$0.68\text{ m}$

Cevap

The length of the pipe is

0.34\text{ m}

The correct answer of

0.34\text{ m}

is obtained by finding the wave speed on the string (

300\text{ m/s}

), computing its 3rd harmonic frequency (

750\text{ Hz}

), and equating this to the 3rd harmonic frequency formula for a closed pipe (

f = \frac{3 v_a}{4 L_p}

Adım Adım Çözüm

Calculate the speed of transverse waves on the stretched string.

v_s = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{360\text{ N}}{4.0 \times 10^{-3}\text{ kg/m}}} = \sqrt{90000} = 300\text{ m/s}

Wave speed on a stretched string depends on tension and linear mass density.

Determine the fundamental frequency and the 2nd overtone frequency of the string.

Fundamental frequency

f_{1,s} = \frac{v_s}{2 L_s} = \frac{300}{2(0.60)} = 250\text{ Hz}

. Second overtone is the 3rd harmonic (

n=3

), so

f = 3 \times 250 = 750\text{ Hz}

For a string fixed at both ends, harmonics are integer multiples of the fundamental, and the 2nd overtone corresponds to

n=3

Relate the resonance frequency to the length of the closed pipe.

First overtone of a closed pipe is its 3rd harmonic (

m=3

f = \frac{3 v_a}{4 L_p} \implies 750 = \frac{3(340)}{4 L_p}

Pipes closed at one end only produce odd harmonics (

1, 3, 5, \dots

), so the 1st overtone is

m=3

Solve for the pipe length

L_p

L_p = \frac{3(340)}{4(750)} = \frac{1020}{3000} = 0.34\text{ m}

Algebraic rearrangement yields the required physical length.

Anahtar Kavram

Harmonics and overtones in vibrating strings and closed air columns under resonance

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