Question

The equation of a wave is $y(x,t)=0.1\sin [\frac{\pi }{3}(10x-30t)-\frac{\pi }{6}]\text{m}$.
Find the acceleration of the particle at $x=0.85\text{m}$ and $t=0.25\text{s}$
[Assume, ${\pi }^2=10$ ; $\uparrow$ denotes positive $y-$direction and $\downarrow$ denotes negative $y-$direction ]

• A. $40{\text{m/s}}^2\uparrow$
• B. $50{\text{m/s}}^2\downarrow$
• C. $40{\text{m/s}}^2\downarrow$
• D. $50{\text{m/s}}^2\uparrow$

Answer 1

The correct option is B $50{\text{m/s}}^2\downarrow$

Given ,
Equation of the wave is
$y(x,t)=0.1\sin [\frac{\pi }{3}(10x-30t)-\frac{\pi }{6}]\text{m}$

Differentiating twice on both sides with respect to time,

$\frac{dy}{dt}=0.1\cos [\frac{\pi }{3}(10x-30t)-\frac{\pi }{6}]\times (-10\pi )$

$\frac{d^{2}y}{d{t}^2}=(-\pi )\sin [\frac{\pi }{3}(10x-30t)-\frac{\pi }{6}]\times (+10\pi )$

So, acceleration of particle $\frac{d^{2}y}{d{t}^2}=-10{\pi }^2\sin [\frac{\pi }{3}(10x-30t)-\frac{\pi }{6}]$

at $x=0.85\text{m}\&t=0.25\text{sec}$ we get ,

$a=-10{\pi }^2\sin [\frac{\pi }{3}(8.5-7.5)-\frac{\pi }{6}]$

$\Rightarrow a=-10{\pi }^2\sin (\frac{\pi }{6})=-50{\text{m/s}}^2$
Thus, option (b) is the correct answer.