Question

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

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

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.

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: \[ \frac{\lambda}{2} = 40 \mathrm{~cm} \]

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.

Key concepts

Fundamental Mode

In the concept of stationary waves, the fundamental mode, also known as the first harmonic, plays a crucial role. Imagine a guitar string vibrates in segments, each separated by points of no motion called nodes. The fundamental mode represents the simplest vibration pattern. In this mode, the string supports just one half-wavelength.

The fundamental mode is a key starting point to understand more complex vibration forms, also known as harmonics. Since it involves only one half-wavelength fitting within the string’s length, it sets the framework for calculating other wave properties.

• One half-wavelength equals the length of the string in this mode.
• This mode determines the pitch or frequency at its most basic level.

Wavelength

Wavelength is a fundamental property of waves and is critical when examining stationary waves. It is defined as the distance between consecutive crests or troughs of the wave. In the context of stationary waves on a string, understanding wavelength helps in finding different modes.

In the fundamental mode, the wavelength is twice the length of the string. This is because one half-wavelength fits perfectly inside the length of the string. So, if the length of the string is 40 cm, the maximum wavelength observed in this mode becomes 80 cm.

• Wavelength in the fundamental mode is always double the string's length.
• This property helps in distinguishing between different harmonics and their respective wavelengths.

Nodes

Nodes are the backbone of understanding stationary waves. They are the points along the wave that remain stationary, even as the wave vibrates. On a string fixed at both ends, nodes naturally occur at these endpoints.

The nodes divide the string into segments, each segment corresponding to a half-wavelength in the case of the fundamental mode. Understanding how nodes function is vital when analyzing how different harmonics form on a wave.

• Nodes are found at both ends of the string in all modes.
• They are points of zero amplitude.

Harmonics

Harmonics refer to the multiple natural frequencies at which a system can vibrate. Apart from the fundamental mode, there are higher modes called harmonics. Each of these corresponds to whole numbers of half-wavelengths fitting within the string length.

For instance, the second harmonic features two half-wavelengths fitting in, the third harmonic involves three, and so on. Understanding harmonics is important for music, engineering, and physics as they represent the additional tones other than the fundamental one.

• Each harmonic number indicates how many half-wavelengths fit within the string’s length.
• Harmonics create richer and more complex sounds.