Correct answer Carries: 4.
Wrong Answer Carries: -1.
A string fixed at both ends has a length of 1.2 m and a wave speed of 72 m/s. What is the frequency of its second harmonic?
For fixed ends: \( v_n = \frac{n v}{2L} \).
Second harmonic (\( n = 2 \)): \( v_2 = \frac{2 \times 72}{2 \times 1.2} = \frac{144}{2.4} = 60 \, \text{Hz} \).
A wave is described by \( y(x, t) = 0.06 \sin (15x - 45t) \), where \( x \) and \( y \) are in meters and \( t \) in seconds. What is the frequency of the wave?
Compare with \( y = a \sin (kx - \omega t) \), \( \omega = 45 \, \text{rad/s} \).
Frequency: \( v = \frac{\omega}{2\pi} = \frac{45}{2\pi} \approx 7.16 \, \text{Hz} \).
A transverse wave travels on a string with tension 250 N and linear mass density 0.05 kg/m. What is the speed of the wave?
Speed: \( v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{250}{0.05}} = \sqrt{5000} \approx 70.71 \, \text{m/s} \).
A closed pipe resonates at 425 Hz with a speed of sound of 340 m/s. What is its length?
Fundamental frequency: \( v_1 = \frac{v}{4L} \).
\( 425 = \frac{340}{4L} \Rightarrow 4L = \frac{340}{425} \Rightarrow 4L = 0.8 \Rightarrow L = 0.2 \, \text{m} \).
A steel rod of density 7800 kg/m³ has a Young’s modulus of 2 × 10¹¹ Pa. What is the speed of a longitudinal wave in the rod?
Speed: \( v = \sqrt{\frac{Y}{\rho}} = \sqrt{\frac{2 \times 10^{11}}{7800}} \approx \sqrt{2.56 \times 10^7} \approx 5060 \, \text{m/s} \).
A pipe open at both ends has a length of 0.55 m and a speed of sound of 330 m/s. What is the frequency of its fourth harmonic?
For open pipe: \( v_n = \frac{n v}{2L} \).
Fourth harmonic (\( n = 4 \)): \( v_4 = \frac{4 \times 330}{2 \times 0.55} = \frac{1320}{1.1} = 1200 \, \text{Hz} \).
What happens to the phase of a transverse wave when it reflects off a free end of a string?
At a free end, the wave reflects without inversion (phase change of 0 radians) because the boundary allows maximum displacement, aligning the reflected wave with the incident wave.
What is the significance of the superposition principle in wave phenomena?
The superposition principle states that the net displacement of a medium is the algebraic sum of individual wave displacements, enabling phenomena like interference and standing waves.
What is the primary factor affecting the speed of sound in a solid rod compared to a gas?
In solids, sound speed depends on Young’s modulus (\( v = \sqrt{\frac{Y}{\rho}} \)), which measures stiffness and is much higher in solids than the bulk modulus in gases, leading to faster sound propagation.
In a progressive wave, what is the key difference between particle displacement and wave propagation direction for a transverse wave?
In a transverse wave, particles oscillate perpendicular to the direction of wave propagation, unlike longitudinal waves where they are parallel.
A wave is described by \( y(x, t) = 0.03 \sin (50x - 200t) \), where \( x \) and \( y \) are in meters and \( t \) in seconds. What is its frequency?
Compare with \( y = a \sin (kx - \omega t) \).
\( \omega = 200 \, \text{rad/s} \).
Frequency: \( v = \frac{\omega}{2\pi} = \frac{200}{2\pi} \approx 31.8 \, \text{Hz} \).
A pipe open at both ends has a length of 0.75 m and a speed of sound of 330 m/s. What is the frequency of its second harmonic?
Second harmonic (\( n = 2 \)): \( v_2 = \frac{2 \times 330}{2 \times 0.75} = \frac{660}{1.5} = 440 \, \text{Hz} \).
A stationary wave is given by \( y = 0.02 \sin (\frac{2\pi x}{7}) \cos (140\pi t) \), where \( x \) and \( y \) are in meters and \( t \) in seconds. What is the wavelength?
Form: \( y = A \sin (kx) \cos (\omega t) \), \( k = \frac{2\pi}{7} \, \text{rad/m} \).
Wavelength: \( \lambda = \frac{2\pi}{k} = \frac{2\pi}{\frac{2\pi}{7}} = 7 \, \text{m} \).
A string of length 1.2 m and mass 0.024 kg has a wave speed of 50 m/s. What is the tension?
\( \mu = \frac{0.024}{1.2} = 0.02 \, \text{kg/m} \).
\( v = \sqrt{\frac{T}{\mu}} \Rightarrow 50 = \sqrt{\frac{T}{0.02}} \Rightarrow 50^2 = \frac{T}{0.02} \).
\( T = 2500 \times 0.02 = 50 \, \text{N} \).
A pipe closed at one end has a length of 0.7 m and resonates at its third harmonic with a speed of sound of 350 m/s. What is the frequency?
For closed pipe: \( v_n = (n + \frac{1}{2}) \frac{v}{2L} \), \( n = 2 \) for third harmonic.
\( v_2 = (2 + \frac{1}{2}) \frac{350}{2 \times 0.7} = 2.5 \times \frac{350}{1.4} = 2.5 \times 250 = 625 \, \text{Hz} \).
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