Question

The equation of a stationary wave is given by \[ y = 5 \sin \frac{\pi}{2} \cos 10\pi t \, {cm} \] The distance between two consecutive nodes (in cm) is:

• A. 5
• B. 2
• C. 8
• D. 1
• E. 6

Hint:

The distance between two consecutive nodes in a stationary wave is half the wavelength.

Answer 1

The Correct Option is B

Step 1: The general equation of a stationary wave is: \[ y = A \sin(kx) \cos(\omega t) \] where:
- \( A \) is the amplitude,
- \( k \) is the wave number,
- \( x \) is the position,
- \( \omega \) is the angular frequency.
In the given equation, we have \( y = 5 \sin \frac{\pi}{2} \cos 10\pi t \). 
Step 2: The wave number \( k \) is related to the wavelength \( \lambda \) by: \[ k = \frac{2\pi}{\lambda} \] From the given equation, \( k = \frac{\pi}{2} \), so the wavelength \( \lambda \) is: \[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{\pi/2} = 4 \, {cm} \] Step 3: The distance between two consecutive nodes is half the wavelength: \[ {Distance between nodes} = \frac{\lambda}{2} = \frac{4}{2} = 2 \, {cm} \] Thus, the distance between two consecutive nodes is 2 cm.