Correct answer Carries: 4.
Wrong Answer Carries: -1.
A wave on a string has an amplitude of 0.06 m and a frequency of 30 Hz. What is the maximum transverse acceleration of a particle?
\( \omega = 2\pi v = 2\pi \times 30 = 60\pi \, \text{rad/s} \).
Max acceleration: \( a_{\text{max}} = a \omega^2 = 0.06 \times (60\pi)^2 \).
\( a_{\text{max}} = 0.06 \times 3600\pi^2 \approx 21330 \, \text{m/s}^2 \).
A sound wave in air has a frequency of 170 Hz and wavelength of 2 m. What is the speed of sound in air?
Speed: \( v = v \lambda = 170 \times 2 = 340 \, \text{m/s} \).
Which condition is essential for the formation of a standing wave in a medium?
Standing waves require waves of equal frequency to interfere, typically from reflection, creating a stationary pattern of nodes and antinodes.
Which factor primarily determines the pitch of a sound wave as perceived by a listener?
Pitch is the auditory perception of frequency. Higher frequency corresponds to higher pitch, while amplitude affects loudness and wavelength relates to speed and frequency.
A string fixed at both ends has a length of 2.5 m and a wave speed of 100 m/s. What is the frequency of its fourth harmonic?
For fixed ends: \( v_n = \frac{n v}{2L} \).
Fourth harmonic (\( n = 4 \)): \( v_4 = \frac{4 \times 100}{2 \times 2.5} = \frac{400}{5} = 80 \, \text{Hz} \).
In a transverse wave, what is the relationship between particle velocity and wave velocity at an antinode of a standing wave?
At an antinode, particle displacement is maximum, oscillating perpendicular to wave propagation. Particle velocity (rate of displacement change) is perpendicular to wave velocity (along the string).
A pipe open at both ends has a length of 0.6 m and a wave speed of 300 m/s. What is the difference in frequency between the third and first harmonics?
\( v_n = \frac{n v}{2L} \).
First harmonic (\( n = 1 \)): \( v_1 = \frac{300}{2 \times 0.6} = 250 \, \text{Hz} \).
Third harmonic (\( n = 3 \)): \( v_3 = \frac{3 \times 300}{2 \times 0.6} = 750 \, \text{Hz} \).
Difference: \( v_3 - v_1 = 750 - 250 = 500 \, \text{Hz} \).
A transverse wave travels on a string with a speed of 20 m/s and a frequency of 10 Hz. What is its wavelength?
Speed: \( v = v \lambda \).
\( \lambda = \frac{v}{v} = \frac{20}{10} = 2 \, \text{m} \).
Why can’t transverse waves propagate through a liquid medium?
Transverse waves require a medium to resist shear deformation (shear modulus). Liquids lack rigidity and cannot sustain shear stress, making transverse wave propagation impossible.
Which property of a gas affects the speed of longitudinal waves more significantly than its density?
Speed of sound in a gas is \( v = \sqrt{\frac{\gamma P}{\rho}} \). The ratio \( \gamma \) (specific heats) influences \( v \) more than \( \rho \) alone, as \( P/\rho \) relates to temperature, but \( \gamma \) varies with molecular structure.
A closed pipe of length 0.3 m has a speed of sound of 330 m/s. What is the frequency of its fifth harmonic?
For closed pipe: \( v_n = (n + \frac{1}{2}) \frac{v}{2L} \), \( n = 4 \) for fifth harmonic.
\( v_4 = (4 + \frac{1}{2}) \frac{330}{2 \times 0.3} = 4.5 \times \frac{330}{0.6} = 4.5 \times 550 = 2475 \, \text{Hz} \).
A stationary wave is given by \( y = 0.05 \sin (\frac{2\pi x}{5}) \cos (100\pi t) \), where \( x \) and \( y \) are in meters and \( t \) in seconds. What is the wavelength?
Form: \( y = A \sin (kx) \cos (\omega t) \), \( k = \frac{2\pi}{5} \, \text{rad/m} \).
Wavelength: \( \lambda = \frac{2\pi}{k} = \frac{2\pi}{\frac{2\pi}{5}} = 5 \, \text{m} \).
A steel wire of length 4 m and mass 0.08 kg is under tension. If a transverse wave takes 0.02 s to travel its length, what is the tension?
Speed: \( v = \frac{\text{length}}{\text{time}} = \frac{4}{0.02} = 200 \, \text{m/s} \).
\( \mu = \frac{0.08}{4} = 0.02 \, \text{kg/m} \).
\( v = \sqrt{\frac{T}{\mu}} \Rightarrow 200 = \sqrt{\frac{T}{0.02}} \Rightarrow 200^2 = \frac{T}{0.02} \).
\( T = 40000 \times 0.02 = 800 \, \text{N} \).
A steel rod of length 2 m and density 7800 kg/m³ has a longitudinal wave speed of 5000 m/s. What is its Young’s modulus?
Speed: \( v = \sqrt{\frac{Y}{\rho}} \).
\( 5000 = \sqrt{\frac{Y}{7800}} \Rightarrow 5000^2 = \frac{Y}{7800} \).
\( Y = 5000^2 \times 7800 = 1.95 \times 10^{11} \, \text{Pa} \).
A pipe open at both ends has a length of 0.57 m and resonates with a source of frequency 900 Hz. What is the harmonic number if the speed of sound is 342 m/s?
\( 900 = \frac{n \times 342}{2 \times 0.57} = \frac{n \times 342}{1.14} \).
\( 900 = n \times 300 \Rightarrow n = \frac{900}{300} = 3 \).
Third harmonic.
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