Question

The equation of a stationary wave along a stretched string is given by \[ y = 5 \sin \left( \frac{\pi x}{3} \right) \cos (40 \pi t) \] Additional Information Here, \(x\) and \(y\) are in cm and \(t\) in seconds. The separation between two adjacent nodes is:

• A. 1.5 cm
• B. 3 cm
• C. 6 cm
• D. 14 cm

Hint:

The separation between two adjacent nodes in a stationary wave is given by the distance between consecutive integer multiples of \( \frac{\pi}{k} \), where \( k \) is the wave number.

Answer 1

The Correct Option is B

We are given the equation of a stationary wave along a stretched string: \[ y = 5 \sin \left( \frac{\pi x}{3} \right) \cos (40 \pi t) \] % Step 1: General Form of a Stationary Wave The general equation of a stationary wave is of the form: \[ y = A \sin(kx) \cos(\omega t) \] where: - \( A \) is the amplitude of the wave, - \( k \) is the wave number, - \( \omega \) is the angular frequency, - \( x \) is the position along the string, - \( t \) is the time. In this case, comparing the given equation with the general form: \[ y = 5 \sin \left( \frac{\pi x}{3} \right) \cos (40 \pi t) \] We identify: - The amplitude \( A = 5 \), - The wave number \( k = \frac{\pi}{3} \), - The angular frequency \( \omega = 40 \pi \). % Step 2: Distance Between Adjacent Nodes In a stationary wave, nodes are points where the displacement is always zero. These occur when the sine term is zero. Thus, the condition for a node is: \[ \sin(kx) = 0 \] The general solution to this is: \[ kx = n\pi, \quad n = 0, 1, 2, 3, \dots \] Solving for \( x \), we get: \[ x = \frac{n\pi}{k} = \frac{n\pi}{\frac{\pi}{3}} = 3n \, \text{cm} \] Thus, the nodes are located at \( x = 0, 3, 6, 9, \dots \). The separation between two adjacent nodes is the distance between two consecutive values of \( x \). For example, the distance between \( x = 0 \) and \( x = 3 \) is: \[ \text{Separation between nodes} = 3 \, \text{cm} \] Thus, the correct answer is 3 cm.