An organ pipe P1, closed at one end vibrating in its first harmonic, and another pipe P2 open at both ends vibrating in its third harmonic are in resonance with a given tuning fork. The ratio of the length of P1 to that of P2 is:
Correct answer: C. 1/6
- A. 8/3
- B. 3/8
- C. 1/6
- D. 1/3
Explanation
An organ pipe closed at one end (P1) vibrating in its first harmonic has a wavelength of 4 times the length of the pipe (since it forms a quarter-wavelength standing wave). An open pipe (P2) vibrating in its third harmonic has a wavelength equal to twice the length of the pipe divided by 3 (since it forms a three-quarter wavelength standing wave). For both pipes to be in resonance with the same tuning fork, their frequencies must be equal. Thus, the ratio of the lengths of P1 to P2 is determined by the ratio of their respective wavelengths, leading to a calculated ratio of 1/6. This results from equating the frequency expressions of the two pipes and solving for the length ratio. The other options do not correctly apply these relationships.v=v/4l1 for first harmonic v=3v/2l2 for third harmonic v/4l1=3v/2l2 l1/l2=1/6
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Progressive waves transfer energy through oscillations, with wavelength, frequency, amplitude and wave speed related by v = fλ; sound speed depends on the medium. Superposition produces interference and stationary waves in strings and organ pipes, while simple harmonic motion describes the oscillation itself and must be distinguished from wave propagation.
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