Correct answer Carries: 4.
Wrong Answer Carries: -1.
A stationary wave on a string fixed at both ends has a frequency of 120 Hz and a wave speed of 48 m/s. What is the length of the string?
Fundamental: \( v_1 = \frac{v}{2L} \).
\( 120 = \frac{48}{2L} \Rightarrow 2L = \frac{48}{120} \Rightarrow 2L = 0.4 \Rightarrow L = 0.2 \, \text{m} \).
A stationary wave is given by \( y = 0.06 \sin (\frac{\pi x}{2}) \cos (120\pi t) \), where \( x \) and \( y \) are in meters and \( t \) in seconds. What is its wavelength?
Form: \( y = A \sin (kx) \cos (\omega t) \), \( k = \frac{\pi}{2} \, \text{rad/m} \).
\( \lambda = \frac{2\pi}{k} = \frac{2\pi}{\frac{\pi}{2}} = 4 \, \text{m} \).
Two waves of frequencies 500 Hz and 504 Hz interfere to produce beats. How many beats are heard in 10 seconds?
Beat frequency: \( v_{\text{beat}} = v_1 - v_2 = 504 - 500 = 4 \, \text{Hz} \).
Beats in 10 s: \( 4 \times 10 = 40 \).
In a standing wave on a string, what is true about the energy at a node?
At a node, displacement is zero, so there is no kinetic energy. The energy is entirely potential due to maximum strain, but no oscillation occurs.
A string of length 1.5 m fixed at both ends has a wave speed of 45 m/s. What is the frequency of its fifth harmonic?
\( v_n = \frac{n v}{2L} \).
Fifth harmonic (\( n = 5 \)): \( v_5 = \frac{5 \times 45}{2 \times 1.5} = \frac{225}{3} = 75 \, \text{Hz} \).
A string of length 2 m and mass 0.04 kg is under a tension of 100 N. What is the speed of a transverse wave on the string?
Linear mass density: \( \mu = \frac{0.04}{2} = 0.02 \, \text{kg/m} \).
Speed: \( v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{100}{0.02}} = \sqrt{5000} \approx 70.71 \, \text{m/s} \).
A pipe open at both ends has a length of 0.45 m and resonates with a source of frequency 1133 Hz. What is the harmonic number if the speed of sound is 340 m/s?
\( 1133 = \frac{n \times 340}{2 \times 0.45} = \frac{n \times 340}{0.9} \).
\( 1133 = n \times 377.78 \Rightarrow n \approx \frac{1133}{377.78} \approx 3 \).
Third harmonic (exact: \( v_3 = 3 \times 340 / 0.9 = 1133.33 \, \text{Hz} \)).
A pipe open at both ends has a length of 0.25 m and a speed of sound of 340 m/s. What is the frequency of its second harmonic?
For open pipe: \( v_n = \frac{n v}{2L} \).
Second harmonic (\( n = 2 \)): \( v_2 = \frac{2 \times 340}{2 \times 0.25} = \frac{680}{0.5} = 1360 \, \text{Hz} \).
Two waves \( y_1 = 4 \sin (6x - 18t) \) and \( y_2 = 4 \sin (6x - 18t + \frac{\pi}{3}) \) interfere. What is the amplitude of the resultant wave?
Amplitude: \( A = 2a \cos \frac{\phi}{2} \), \( a = 4 \, \text{m} \), \( \phi = \frac{\pi}{3} \).
\( A = 2 \times 4 \cos \frac{\pi}{6} = 8 \times \frac{\sqrt{3}}{2} \approx 6.93 \, \text{m} \).
A longitudinal wave in a medium has a speed of 1450 m/s and a density of 1100 kg/m³. What is the bulk modulus of the medium?
Speed: \( v = \sqrt{\frac{B}{\rho}} \).
\( 1450 = \sqrt{\frac{B}{1100}} \Rightarrow 1450^2 = \frac{B}{1100} \).
\( B = 1450^2 \times 1100 = 2.31 \times 10^9 \, \text{Pa} \).
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