Question

The equation for a wave travelling in $x$ direction on a string is given as
$y=10\sin (\pi t-0.5x)$, where $y$ and $x$ are in $\text{m}$ and time$\left(t\right)$ in $\text{sec}$. Find the acceleration of a particle at $x=\pi \text{m}$ at time $t=1\text{sec}$.

• A. $-10\pi {\text{m/s}}^2$
• B. $-10{\pi }^2{\text{m/s}}^2$
• C. $5\pi {\text{m/s}}^2$
• D. $-5\pi {\text{m/s}}^2$

Answer 1

The correct option is B $-10{\pi }^2{\text{m/s}}^2$

We know,
Velocity of particle $v_P=\frac{\partial y}{\partial t}=10\pi \cos (\pi t-0.5x)$
Similarly, acceleration of particle can be obtained by partial differentiation of velocity w.r.t time (keeping $x$ constant).
$\Rightarrow a_P=\frac{\partial v_P}{\partial t}$
$\Rightarrow a_P=\pi \times -10\pi \sin (\pi t-0.5x)$
$\Rightarrow a_P=-10{\pi }^2\sin (\pi t-0.5x)$
substituting $x=\pi \text{m}\&t=1\text{sec}$ in above equation we get,
$a_P=-10{\pi }^2\sin (\pi -0.5\pi )$
$a_P=-10{\pi }^2\sin \frac{\pi }{2}$
$\therefore a_P=-10{\pi }^2{\text{m/s}}^2$