Correct answer Carries: 4.
Wrong Answer Carries: -1.
Which property of a medium is most critical for the propagation of longitudinal waves through a gas?
Longitudinal waves in a gas rely on compressibility, quantified by the bulk modulus, which provides the restoring force for pressure oscillations, unlike shear modulus needed for transverse waves.
Two waves of frequencies 510 Hz and 514 Hz interfere. How many beats are heard in 15 seconds?
Beat frequency: \( v_{\text{beat}} = 514 - 510 = 4 \, \text{Hz} \).
Beats in 15 s: \( 4 \times 15 = 60 \).
A stationary wave on a string fixed at both ends has a frequency of 150 Hz and a wave speed of 45 m/s. What is the wavelength?
Wavelength: \( \lambda = \frac{v}{v} = \frac{45}{150} = 0.3 \, \text{m} \).
A transverse wave on a string has a tension of 180 N and a linear mass density of 0.03 kg/m. What is the wavelength if the frequency is 20 Hz?
Speed: \( v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{180}{0.03}} = \sqrt{6000} \approx 77.46 \, \text{m/s} \).
Wavelength: \( \lambda = \frac{v}{v} = \frac{77.46}{20} \approx 3.87 \, \text{m} \).
A pipe open at both ends has a length of 0.34 m and resonates with a source of frequency 1000 Hz. What is the harmonic number if the speed of sound is 340 m/s?
\( v_n = \frac{n v}{2L} \).
\( 1000 = \frac{n \times 340}{2 \times 0.34} = \frac{n \times 340}{0.68} \).
\( 1000 = n \times 500 \Rightarrow n = \frac{1000}{500} = 2 \).
Second harmonic.
A string of length 4 m and mass 0.08 kg is under a tension of 200 N. How long does a transverse pulse take to travel its length?
Linear mass density: \( \mu = \frac{0.08}{4} = 0.02 \, \text{kg/m} \).
Speed: \( v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{200}{0.02}} = \sqrt{10000} = 100 \, \text{m/s} \).
Time: \( t = \frac{\text{length}}{v} = \frac{4}{100} = 0.04 \, \text{s} \).
Two waves \( y_1 = 4 \sin (5x - 10t) \) and \( y_2 = 4 \sin (5x - 10t + \frac{\pi}{6}) \) interfere. What is the amplitude of the resultant wave?
Amplitude: \( A = 2a \cos \frac{\phi}{2} \), \( a = 4 \, \text{m} \), \( \phi = \frac{\pi}{6} \).
\( A = 2 \times 4 \cos \frac{\pi}{12} \approx 8 \times 0.966 = 7.73 \, \text{m} \).
A steel rod of length 1.5 m has a fundamental frequency of longitudinal vibrations of 2 kHz. What is the speed of sound in the rod?
For rod clamped at middle, fundamental: \( v_1 = \frac{v}{2L} \).
\( 2000 = \frac{v}{2 \times 1.5} \Rightarrow v = 2000 \times 3 = 6000 \, \text{m/s} \).
A pipe open at both ends has a length of 0.68 m and resonates with a source of frequency 750 Hz. What is the harmonic number if the speed of sound is 340 m/s?
\( 750 = \frac{n \times 340}{2 \times 0.68} = \frac{n \times 340}{1.36} \).
\( 750 = n \times 250 \Rightarrow n = \frac{750}{250} = 3 \).
Third harmonic.
A string of length 3 m and mass 0.06 kg is under a tension of 150 N. What is the speed of a transverse wave on the string?
Linear mass density: \( \mu = \frac{0.06}{3} = 0.02 \, \text{kg/m} \).
Speed: \( v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{150}{0.02}} = \sqrt{7500} \approx 86.6 \, \text{m/s} \).
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