Question

A particle on a string undergoes a transverse wave motion given by: \[ y = 5 \sin \left( 4 \pi t - \frac{\pi x}{2} \right) \] (All quantities in SI units.) How much time does a particle at \( x = 0 \) take to go from mean position to extreme (maximum displacement) for the first time?

• A. \( \frac{1}{4} \) S
• B. \( \frac{1}{8} \) S
• C. \( \frac{1}{2} \) S
• D. \( \frac{1}{16} \) S

Hint:

The time for a particle to move from the mean position to extreme displacement in a sinusoidal wave is \( \frac{1}{8} \) of the time period, since it reaches the extreme for the first time at \( \frac{\pi}{2} \) radians.

Answer 1

The Correct Option is B

The equation of the transverse wave is given by: \[ y = 5 \sin \left( 4 \pi t - \frac{\pi x}{2} \right) \] At \( x = 0 \), the equation simplifies to: \[ y = 5 \sin(4 \pi t) \] Now, the maximum displacement occurs when \( \sin(4 \pi t) = \pm 1 \). For the first time, the particle moves from the mean position (where \( y = 0 \)) to the extreme (where \( y = 5 \)). At \( t = 0 \), the particle is at the mean position (\( y = 0 \)). The particle will reach the extreme for the first time when: \[ \sin(4 \pi t) = 1 \] This occurs when: \[ 4 \pi t = \frac{\pi}{2} \] Solving for \( t \): \[ t = \frac{1}{8} \, \text{seconds} \] Thus, the time taken for the particle to move from the mean position to extreme displacement is \( \frac{1}{8} \) seconds.