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 \( y = 5 \sin \left( 4 \pi t - \frac{\pi x}{2} \right) \). When \( x = 0 \), the equation becomes \( y = 5 \sin(4 \pi t) \). The maximum displacement occurs when \( \sin(4 \pi t) = \pm 1 \). The particle moves from the mean position (\( y = 0 \)) to an extreme position (\( y = 5 \)) for the first time. At \( t = 0 \), the particle is at the mean position (\( y = 0 \)). The particle reaches the extreme for the first time when \( \sin(4 \pi t) = 1 \). This condition is met when \( 4 \pi t = \frac{\pi}{2} \). Solving for \( t \) yields \( t = \frac{1}{8} \) seconds. Therefore, the time required for the particle to move from the mean position to the extreme displacement is \( \frac{1}{8} \) seconds.