Correct answer Carries: 4.
Wrong Answer Carries: -1.
Why does a standing wave in a pipe closed at one end produce only odd harmonics?
The boundary condition (node at the closed end, antinode at the open end) requires the wavelength to fit \( L = (2n - 1) \frac{\lambda}{4} \), allowing only odd multiples of the fundamental.
A transverse wave travels on a string with tension 400 N and linear mass density 0.1 kg/m. What is the speed of the wave?
Speed: \( v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{400}{0.1}} = \sqrt{4000} \approx 63.25 \, \text{m/s} \).
A transverse wave on a string has a tension of 120 N and a linear mass density of 0.015 kg/m. What is the wavelength if the frequency is 50 Hz?
Speed: \( v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{120}{0.015}} = \sqrt{8000} \approx 89.44 \, \text{m/s} \).
Wavelength: \( \lambda = \frac{v}{v} = \frac{89.44}{50} \approx 1.79 \, \text{m} \).
What happens to a transverse wave when it reflects off a fixed end of a string?
At a fixed end, the displacement must be zero, requiring the reflected wave to be out of phase by \( \pi \) radians (inverted) to cancel the incident wave at the boundary.
Two strings produce beats of 4 Hz. If one has a frequency of 256 Hz and the tension in the other is increased, the beat frequency increases to 6 Hz. What was the original frequency of the second string?
Let original frequency of second string = \( v_2 \).
\( |256 - v_2| = 4 \Rightarrow v_2 = 252 \, \text{Hz or } 260 \, \text{Hz} \).
Tension increase raises frequency. If \( v_2 = 252 \), new \( v_2 > 252 \), beat = \( 256 - v_2’ < 4 \), contradicts.
If \( v_2 = 260 \), new \( v_2 > 260 \), beat = \( v_2’ - 256 = 6 \Rightarrow v_2’ = 262 \, \text{Hz} \), consistent.
So, \( v_2 = 260 \, \text{Hz} \).
Which phenomenon explains the periodic waxing and waning of sound intensity when two sound waves of slightly different frequencies interfere?
Beats occur when two waves of slightly different frequencies superpose, causing constructive and destructive interference periodically, resulting in alternating loudness (waxing and waning).
A pipe closed at one end has a length of 0.32 m and resonates at its fundamental frequency with a speed of sound of 320 m/s. What is the frequency?
Fundamental: \( v_1 = \frac{v}{4L} \).
\( v_1 = \frac{320}{4 \times 0.32} = \frac{320}{1.28} = 250 \, \text{Hz} \).
A transverse wave on a string has a speed of 25 m/s and a frequency of 20 Hz. What is its wavelength?
Speed: \( v = v \lambda \).
Wavelength: \( \lambda = \frac{v}{v} = \frac{25}{20} = 1.25 \, \text{m} \).
In a pipe closed at one end, which harmonics are absent compared to a pipe open at both ends?
A pipe closed at one end has frequencies \( v_n = (2n - 1) \frac{v}{4L} \), producing only odd harmonics (1, 3, 5, ...). A pipe open at both ends has all harmonics (1, 2, 3, ...), so even harmonics are absent in the closed pipe.
A wave on a string has an amplitude of 0.025 m and a frequency of 50 Hz. What is the maximum transverse speed of a particle?
\( \omega = 2\pi v = 2\pi \times 50 = 100\pi \, \text{rad/s} \).
Max speed: \( v_{\text{max}} = a \omega = 0.025 \times 100\pi \approx 7.85 \, \text{m/s} \).
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