If an open organ pipe is sounded with a tuning fork having...

If an open organ pipe is sounded with a tuning fork having frequency $256 \mathrm{~Hz}$, resonance occurs at $35 \mathrm{~cm}$ and $105 \mathrm{~cm}$, the velocity of sound is (a) $360 \mathrm{~m} / \mathrm{s}$ (b) $512 \mathrm{~m} / \mathrm{s}$ (c) $524 \mathrm{~m} / \mathrm{s}$ (d) all of these

If an open organ pipe is sounded with a tuning fork having frequency $256 \mathrm{~Hz}$, resonance occurs at $35 \mathrm{~cm}$ and $105 \mathrm{~cm}$, the velocity of sound is
(a) $360 \mathrm{~m} / \mathrm{s}$
(b) $512 \mathrm{~m} / \mathrm{s}$
(c) $524 \mathrm{~m} / \mathrm{s}$
(d) all of these

Key Concepts

Successive Resonance Positions

In systems like an open organ pipe, consecutive resonances occur when the pipe length increases by half a wavelength (?/2). By measuring the difference between successive resonance positions, one can deduce the value of ?/2 and thus determine the full wavelength, which is critical for calculating the velocity of sound.

Open Organ Pipe Resonance

In an open organ pipe, both ends are open, so the pressure at each end must be constant, resulting in a pressure node at each end. The resonance condition is met when the length of the pipe is an integer multiple of half the wavelength (L = n(?/2) for n = 1, 2, 3, …). This principle helps in identifying the wavelengths that can exist in the pipe for a given mode of vibration.

Standing Waves

Standing waves are produced when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. This interference creates a pattern of nodes, where displacement is zero, and antinodes, where displacement is maximum. The formation of these standing waves in musical instruments, such as organ pipes, explains the occurrence of resonant frequencies.

Frequency-Wavelength-Velocity Relationship

The fundamental relationship between frequency, wavelength, and velocity is expressed by the equation v = f?, where v is the speed of sound, f is its frequency, and ? is its wavelength. This equation is essential for calculating one of these parameters if the other two are known.

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