Correct answer Carries: 4.
Wrong Answer Carries: -1.
A transverse wave on a string is described by \( y(x, t) = 0.04 \sin (60x - 15t) \), where \( x \) and \( y \) are in meters and \( t \) in seconds. What is the speed of the wave?
Compare with \( y(x, t) = a \sin (kx - \omega t) \).
Here, \( k = 60 \, \text{rad/m} \), \( \omega = 15 \, \text{rad/s} \).
Speed of wave: \( v = \frac{\omega}{k} = \frac{15}{60} = 0.25 \, \text{m/s} \).
A pipe open at both ends has a length of 0.65 m and a speed of sound of 340 m/s. What is the frequency of its third harmonic?
For open pipe: \( v_n = \frac{n v}{2L} \).
Third harmonic (\( n = 3 \)): \( v_3 = \frac{3 \times 340}{2 \times 0.65} = \frac{1020}{1.3} \approx 784.62 \, \text{Hz} \).
A transverse wave on a string has a tension of 150 N and a linear mass density of 0.02 kg/m. What is the frequency if the wavelength is 2.5 m?
Speed: \( v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{150}{0.02}} = \sqrt{7500} \approx 86.6 \, \text{m/s} \).
Frequency: \( v = \frac{v}{\lambda} = \frac{86.6}{2.5} \approx 34.64 \, \text{Hz} \).
Which statement best describes the energy in a standing wave?
In a standing wave, energy oscillates between kinetic (at antinodes) and potential (at nodes) forms but does not propagate, remaining trapped within the medium.
Two waves of frequencies 520 Hz and 526 Hz interfere. How many beats are heard in 10 seconds?
Beat frequency: \( v_{\text{beat}} = 526 - 520 = 6 \, \text{Hz} \).
Beats in 10 s: \( 6 \times 10 = 60 \).
A copper rod of density 8960 kg/m³ has a speed of sound of 3600 m/s. What is its Young’s modulus?
Speed: \( v = \sqrt{\frac{Y}{\rho}} \).
\( 3600 = \sqrt{\frac{Y}{8960}} \Rightarrow 3600^2 = \frac{Y}{8960} \).
\( Y = 3600^2 \times 8960 = 1.16 \times 10^{11} \, \text{Pa} \).
For a standing wave in a pipe open at both ends, which statement is true about the harmonic frequencies?
In a pipe open at both ends, the frequencies are \( v_n = n \frac{v}{2L} \), where \( n = 1, 2, 3, ... \), meaning all harmonics (integer multiples of the fundamental) are present.
A longitudinal wave in a medium has a speed of 1500 m/s and a bulk modulus of 2.25 × 10⁹ Pa. What is the density of the medium?
Speed: \( v = \sqrt{\frac{B}{\rho}} \).
\( 1500 = \sqrt{\frac{2.25 \times 10^9}{\rho}} \Rightarrow 1500^2 = \frac{2.25 \times 10^9}{\rho} \).
\( \rho = \frac{2.25 \times 10^9}{1500^2} = \frac{2.25 \times 10^9}{2.25 \times 10^6} = 1000 \, \text{kg/m}^3 \).
A wave \( y = 0.07 \sin (25x - 50t) \) reflects at a rigid boundary. What is the equation of the reflected wave?
At rigid boundary, phase changes by \( \pi \).
Incident: \( y_i = 0.07 \sin (25x - 50t) \).
Reflected: \( y_r = -0.07 \sin (25x + 50t) \).
What happens to the amplitude of two identical waves undergoing destructive interference when their phase difference is \( \pi \) radians?
For two identical waves with phase difference \( \phi = \pi \), the resultant amplitude is \( A = 2a \cos(\phi/2) = 2a \cos(\pi/2) = 0 \), leading to complete cancellation.
A pipe closed at one end has a length of 0.36 m and a speed of sound of 324 m/s. What is the frequency of its third harmonic?
For closed pipe: \( v_n = (n + \frac{1}{2}) \frac{v}{2L} \), \( n = 2 \) for third harmonic.
\( v_2 = (2 + \frac{1}{2}) \frac{324}{2 \times 0.36} = 2.5 \times \frac{324}{0.72} = 2.5 \times 450 = 1125 \, \text{Hz} \).
A pipe closed at one end has a length of 0.68 m and a speed of sound of 340 m/s. What is the frequency of its third harmonic?
\( v_2 = (2 + \frac{1}{2}) \frac{340}{2 \times 0.68} = 2.5 \times \frac{340}{1.36} = 2.5 \times 250 = 625 \, \text{Hz} \).
What happens to the speed of a transverse wave on a string if the tension is doubled while keeping the linear mass density constant?
The speed of a transverse wave is \( v = \sqrt{\frac{T}{\mu}} \). If tension \( T \) doubles, \( v' = \sqrt{\frac{2T}{\mu}} = \sqrt{2} \times v \), increasing by a factor of \( \sqrt{2} \) (approximately 1.414).
A string of mass 0.05 kg and length 5 m is under tension. If the speed of a transverse wave is 100 m/s, what is the tension?
\( \mu = \frac{\text{mass}}{\text{length}} = \frac{0.05}{5} = 0.01 \, \text{kg/m} \).
\( v = \sqrt{\frac{T}{\mu}} \Rightarrow 100 = \sqrt{\frac{T}{0.01}} \).
\( 100^2 = \frac{T}{0.01} \Rightarrow T = 10000 \times 0.01 = 100 \, \text{N} \).
A string of length 2.1 m fixed at both ends has a wave speed of 63 m/s. What is the frequency of its fifth harmonic?
\( v_n = \frac{n v}{2L} \).
Fifth harmonic (\( n = 5 \)): \( v_5 = \frac{5 \times 63}{2 \times 2.1} = \frac{315}{4.2} = 75 \, \text{Hz} \).
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