Question

A string of length \(1.5 \mathrm{~m}\) with its two ends clamped is vibrating in fundamental mode. Amplitude at the centre of the string is \(4 \mathrm{~mm}\). Minimum distance between the two points having amplitude \(2 \mathrm{~mm}\) is : (1) \(1 \mathrm{~m}\) (2) \(75 \mathrm{~cm}\) (3) \(60 \mathrm{~cm}\) (4) \(50 \mathrm{~cm}\)

Short answer

1 m

Step-by-step solution

Understand the Fundamental Mode

In the fundamental mode of vibration for a string with both ends clamped, there is one antinode at the center and nodes at the two ends. The length of the string \( L \) is equal to half the wavelength \( \frac{\lambda}{2} \). Thus, \( \lambda = 2L = 2 \times 1.5 \text{ m} = 3 \text{ m} \).

Identify Amplitudes and Distances

Amplitude in a standing wave depends on the position along the string. The maximum amplitude at the antinode (center of the string) is 4 mm.

Use Sine Function for Amplitude

The amplitude \( A(x) \) at a distance \( x \) from a node is given by the function: \[ A(x) = A_{\text{max}} \sin \left( \frac{\pi x}{L} \right) \] where \( A_{\text{max}} \) is the maximum amplitude (4 mm).

Set Up the Equation for Given Amplitude

For the points where the amplitude is 2 mm, use \[ A(x) = 2 = 4 \sin \left( \frac{\pi x}{1.5} \right) \] Simplifying, \[ \sin \left( \frac{\pi x}{1.5} \right) = \frac{1}{2} \]

Solve for \( x \)

\( \sin \left( \frac{\pi x}{1.5} \right) = \frac{1}{2} \) has solutions \( \frac{\pi x}{1.5} = \frac{\pi}{6} \) and \( \frac{\pi x}{1.5} = \frac{5\pi}{6} \). Thus, \( x = \frac{1.5}{6} = 0.25 \text{ m} \) and \( x = 1.25 \text{ m} \).

Calculate the Minimum Distance

The distance between these two points is \( x_2 - x_1 = 1.25 \text{ m} - 0.25 \text{ m} = 1.00 \text{ m} \).

Key concepts

Standing Waves

Standing waves are a fascinating phenomenon seen in many physical systems, such as musical instruments and vibrating strings. They occur when two waves of the same frequency and amplitude travel in opposite directions and interfere with one another. This interference results in a wave pattern that appears to be standing still. In the case of a string clamped at both ends, standing waves are produced as the wave reflects back and forth along the string, forming fixed points called nodes and antinodes. Nodes are points where the wave has zero amplitude and remain stationary, while antinodes are points where the wave reaches maximum amplitude and undergoes maximum oscillation.

Amplitude

Amplitude is a key property of waves that describes the maximum displacement of points on the wave from their rest position. It determines the energy and intensity of the wave. In a standing wave on a vibrating string, the amplitude varies along the length of the string. The maximum amplitude, known as the antinode, is located at the center when the string vibrates in its fundamental mode. In the given exercise, the center of the string has an amplitude of 4 mm, while points along the string will have lower amplitudes depending on their position relative to the nodes and antinodes.

Nodal Points

Nodal points, or simply nodes, are specific points on a standing wave where there is no movement. This means the amplitude is zero due to the destructive interference of the two waves traveling in opposite directions. On a string clamped at both ends, these nodes appear at the endpoints and at various intervals along the string. The distance between two consecutive nodes is half the wavelength of the wave. When dealing with the fundamental mode of a vibrating string, the nodes are found at the two ends of the string and any additional nodes that form depend on the harmonic mode of the vibration.

Wavelength

Wavelength is the distance between successive crests, troughs, or identical points of the wave. It plays a critical role in determining the frequency and speed of wave propagation. For a string clamped at both ends vibrating in its fundamental mode, the wavelength is twice the length of the string. In the exercise, the string's length is 1.5 meters, so the wavelength is 3 meters. By understanding the wavelength, we can predict where nodes and antinodes will form along the string. Moreover, wavelength helps us solve for points with specific amplitudes by using mathematical relationships, such as the sine function, to describe wave behavior.