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Chapter 10: Problem 1499

The length of a string tied across two rigid supports is \(40 \mathrm{~cm}\). The maximum wavelength of a stationary wave that can be produced in it is \(\ldots \ldots \ldots \mathrm{cm}\). (A) 20 (B) 40 (C) 80 (D) 120

Short Answer

Expert verified
The maximum wavelength of a stationary wave on a string of length 40 cm corresponds to the fundamental mode of vibration. In this mode, there is one half-wavelength across the length of the string. Using the relationship \(\frac{\lambda}{2}= L\), we find the maximum wavelength to be \(\lambda = 2 \times 40 \mathrm{~cm} = 80 \mathrm{~cm}\). Therefore, the correct answer is (C) 80.

Step by step solution

01

Understand the properties of a stationary wave

A stationary wave is the interference pattern of two continuous traveling waves of the same frequency and amplitude, moving in opposite directions. In this case, the string is fixed at both ends, so the stationary wave will have nodes at the supports. The modes of vibration correspond to the number of half-wavelengths that can fit within the length of the string. The fundamental mode (also known as the first harmonic) has one half-wavelength on the string, the second harmonic has two half-wavelengths, and so on. The maximum wavelength of a stationary wave corresponds to the fundamental mode of vibration.
02

Use the relationship between the half-wavelength and the length of the string

In the fundamental mode, there is one half-wavelength across the length of the string. Let's denote the half-wavelength as \( \frac{\lambda}{2} \), and L as the length of the string: \[ \frac{\lambda}{2}= L \] Now, substitute the given length of the string, 40 cm, into the equation: \[ \frac{\lambda}{2} = 40 \mathrm{~cm} \]
03

Solve for the maximum wavelength

To find the maximum wavelength, λ, of the stationary wave, we need to multiply both sides of the equation by 2: \[ \lambda = 2 \times 40 \mathrm{~cm} \] Calculating the result: \[ \lambda = 80 \mathrm{~cm} \] The maximum wavelength of a stationary wave that can be produced on the given string is 80 cm. So, the correct answer is (C) 80.

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