Correct answer Carries: 4.
Wrong Answer Carries: -1.
In a transverse wave traveling along a stretched string, which property of the medium primarily determines the speed of the wave?
The speed of a transverse wave on a string is given by \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is tension and \( \mu \) is linear mass density. Tension is a property imposed on the medium, and it primarily influences the speed by providing the restoring force, while \( \mu \) relates to inertia.
Two waves \( y_1 = 5 \sin (6x - 12t) \) and \( y_2 = 5 \sin (6x - 12t + \frac{\pi}{3}) \) interfere. What is the amplitude of the resultant wave?
Amplitude: \( A = 2a \cos \frac{\phi}{2} \), \( a = 5 \, \text{m} \), \( \phi = \frac{\pi}{3} \).
\( A = 2 \times 5 \cos \frac{\pi}{6} = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3} \approx 8.66 \, \text{m} \).
A wave is described by \( y(x, t) = 0.03 \sin (35x - 70t) \), where \( x \) and \( y \) are in meters and \( t \) in seconds. What is the speed of the wave?
Compare with \( y = a \sin (kx - \omega t) \), \( k = 35 \, \text{rad/m} \), \( \omega = 70 \, \text{rad/s} \).
Speed: \( v = \frac{\omega}{k} = \frac{70}{35} = 2 \, \text{m/s} \).
A copper rod has a density of 8900 kg/m³ and a longitudinal wave speed of 3560 m/s. What is its Young’s modulus?
\( v = \sqrt{\frac{Y}{\rho}} \).
\( 3560 = \sqrt{\frac{Y}{8900}} \Rightarrow 3560^2 = \frac{Y}{8900} \).
\( Y = 3560^2 \times 8900 \approx 1.13 \times 10^{11} \, \text{Pa} \).
What is the relationship between the wavelengths of the fundamental mode and the second harmonic in a string fixed at both ends?
For a string fixed at both ends, fundamental wavelength is \( \lambda_1 = 2L \), and second harmonic is \( \lambda_2 = L \). Thus, \( \lambda_1 = 2 \lambda_2 \), or \( \lambda_2 = \lambda_1 / 2 \).
A wave \( y = 0.06 \sin (18x - 54t) \) reflects at an open boundary. What is the equation of the reflected wave?
At open boundary, no phase change.
Incident: \( y_i = 0.06 \sin (18x - 54t) \).
Reflected: \( y_r = 0.06 \sin (18x + 54t) \).
A string of mass 0.06 kg and length 3 m is under a tension of 150 N. How long does it take a transverse pulse to travel its length?
Linear mass density: \( \mu = \frac{0.06}{3} = 0.02 \, \text{kg/m} \).
Speed: \( v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{150}{0.02}} = \sqrt{7500} \approx 86.6 \, \text{m/s} \).
Time: \( t = \frac{\text{length}}{v} = \frac{3}{86.6} \approx 0.035 \, \text{s} \).
What is the condition for maximum constructive interference between two waves?
Maximum constructive interference occurs when the phase difference is zero or a multiple of \( 2\pi \), aligning crests with crests for maximum amplitude.
A string of length 3.2 m and mass 0.064 kg has a fundamental frequency of 25 Hz. What is the tension in the string?
\( \mu = \frac{0.064}{3.2} = 0.02 \, \text{kg/m} \).
\( v_1 = \frac{v}{2L} \Rightarrow 25 = \frac{v}{2 \times 3.2} \Rightarrow v = 25 \times 6.4 = 160 \, \text{m/s} \).
\( v = \sqrt{\frac{T}{\mu}} \Rightarrow 160 = \sqrt{\frac{T}{0.02}} \Rightarrow 160^2 = \frac{T}{0.02} \).
\( T = 25600 \times 0.02 = 512 \, \text{N} \).
A string fixed at both ends has a length of 1.6 m and a wave speed of 80 m/s. What is the frequency of its first harmonic?
For fixed ends: \( v_n = \frac{n v}{2L} \).
First harmonic (\( n = 1 \)): \( v_1 = \frac{80}{2 \times 1.6} = \frac{80}{3.2} = 25 \, \text{Hz} \).
What causes the intensity of a sound wave to decrease as it travels through a medium?
Energy dissipation occurs due to absorption and scattering in the medium, reducing intensity as the wave spreads and loses energy to internal friction or obstacles.
Which condition must be satisfied for two waves to form a standing wave pattern?
Standing waves form when two waves of the same frequency travel in opposite directions and interfere, typically due to reflection, creating nodes and antinodes.
A pipe open at both ends has a length of 0.6 m and resonates with a source of frequency 850 Hz. What is the harmonic number if the speed of sound is 340 m/s?
\( v_n = \frac{n v}{2L} \).
\( 850 = \frac{n \times 340}{2 \times 0.6} = \frac{n \times 340}{1.2} \).
\( 850 = n \times 283.33 \Rightarrow n \approx \frac{850}{283.33} \approx 3 \).
Third harmonic (exact: \( v_3 = 3 \times 340 / 1.2 = 850 \, \text{Hz} \)).
The bulk modulus of water is 2.2 × 10⁹ Pa and its density is 1000 kg/m³. What is the speed of sound in water?
Speed: \( v = \sqrt{\frac{B}{\rho}} = \sqrt{\frac{2.2 \times 10^9}{1000}} = \sqrt{2.2 \times 10^6} \approx 1483 \, \text{m/s} \).
A transverse wave on a string has a speed of 15 m/s and a period of 0.05 s. What is its wavelength?
Frequency: \( v = \frac{1}{T} = \frac{1}{0.05} = 20 \, \text{Hz} \).
Wavelength: \( \lambda = \frac{v}{v} = \frac{15}{20} = 0.75 \, \text{m} \).
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