When the speed of sound in air is 330 m / s, the shortest air column, closed at one end that wi

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When the speed of sound in air is 330 m / s, the shortest...

When the speed of sound in air is $330 \mathrm{~m} / \mathrm{s}$, the shortest air column, closed at one end that will respond to a tuning for $k$ with a frequency of $440 \mathrm{vibs} / \mathrm{s}$ has a length of (approximately). (A) $19 \mathrm{~cm}$ (B) $33 \mathrm{~cm}$ (C) $38 \mathrm{~cm}$ (D) $67 \mathrm{~cm}$

When the speed of sound in air is $330 \mathrm{~m} / \mathrm{s}$, the shortest air column, closed at one end that will respond to a tuning for $k$ with a frequency of $440 \mathrm{vibs} / \mathrm{s}$ has a length of (approximately).
(A) $19 \mathrm{~cm}$
(B) $33 \mathrm{~cm}$
(C) $38 \mathrm{~cm}$
(D) $67 \mathrm{~cm}$

Key Concepts

Resonance in Air Columns

In air columns, resonance occurs when sound waves reflect off the boundaries and interfere constructively. The specific modes of resonance depend on whether the column is open at both ends or closed at one end, leading to different patterns of nodes and antinodes.

Standing Waves

Standing waves are the result of interference between two waves traveling in opposite directions. In resonant systems, such as strings or air columns, certain frequencies produce standing waves, which are characterized by nodes (points of no displacement) and antinodes (points of maximum displacement).

Quarter Wavelength Resonance

For an air column closed at one end, the fundamental mode of resonance occurs when the length of the column is one quarter of the wavelength of the sound. This configuration creates a node at the closed end and an antinode at the open end, setting up a standing wave pattern.

Frequency

Frequency refers to the number of oscillations of a wave per unit time, measured in hertz (Hz). It is a key parameter in wave mechanics that determines the pitch of a sound when it comes to acoustics.

Speed of Sound

The speed of sound is the rate at which sound waves propagate through a medium. It is a fundamental property that depends on the medium's characteristics, such as temperature and density. In many physics problems, the speed of sound is used to relate the frequency and wavelength of sound waves.

Wavelength and Frequency Relationship

The relationship between wavelength, frequency, and the speed of sound is given by the equation: speed = frequency × wavelength. This fundamental principle allows one to calculate any one of these quantities provided the other two are known.

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