Concept explainers
(a)
The wavelength of the fundamental mode of vibration.
(a)
Answer to Problem 81P
The wavelength of the fundamental mode is
Explanation of Solution
The length of the string is
Write the expression for wavelength of the fundamental mode
Here,
Substitute
Thus, the wavelength of the fundamental mode is
(b)
The speed of the wave.
(b)
Answer to Problem 81P
The speed of the wave is
Explanation of Solution
Write the expression for speed of the wave
Here,
Substitute
Thus, the speed of the wave is
(c)
The linear mass density of the string.
(c)
Answer to Problem 81P
The linear mass density of the string is
Explanation of Solution
Write the expression for velocity of wave propagation on the string
Here,
Rearrange for
Substitute
Thus, the linear mass density of the string is
(d)
The maximum speed of a point on the vibrating string.
(d)
Answer to Problem 81P
The maximum speed of a point on the vibrating string
Explanation of Solution
The maximum amplitude on the string is
Write the expression for maximum speed
Here,
Substitute for
Here,
Substitute
Thus, the linear mass density of the string is
(e)
The frequency of the sound wave.
(e)
Answer to Problem 81P
The frequency of the sound wave in air is
Explanation of Solution
The frequency of the vibrating string is
The vibrating string puts the air molecules around it into vibratory motion. The frequency of vibration of the air molecules is same as the frequency of the vibrating string.
Hence the frequency of the sound wave in air is
(f)
The wavelength of sound in air medium at
(f)
Answer to Problem 81P
The wavelength of sound at
Explanation of Solution
The air temperature in kelvin is
Rearrange (II)
Here,
Write the expression for temperature dependence of velocity of sound
Here,
Substitute
Substitute
Thus, the wavelength of sound at
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