Question

The equation of a tranverse wave propagating through a medium is given by $y=10\sin (\pi t+0.5x)$. Choose the correct statement for the velocity of particle present at $x=0$.

• A. At $t=0$, velocity of particle is $0\text{m/s}$
• B. At $t=0.5\text{sec}$, velocity of particle is $\pi \text{m/s}$
• C. At $t=3\text{sec}$, velocity of particle is $10\pi \text{m/s}$
• D. At $t=0.5\text{sec}$, velocity of particle is $0\text{m/s}$

Answer 1

The correct option is D At $t=0.5\text{sec}$, velocity of particle is $0\text{m/s}$

The displacement equation for the wave is,
$y=10\sin (\pi t+0.5x)$
Differentiating above equation partially w.r.t time gives velocity of particle at a fixed position.
$\Rightarrow v_P=\frac{\partial y}{\partial t}$
$\Rightarrow v_P=\pi \times 10\cos (\pi t+0.5x)$
Substituting $x=0$ in above equation to get particle velocity at $x=0$,
$\Rightarrow v_P=\pi \times 10\cos (\pi t+0)$
$\Rightarrow v_P=10\pi \cos \pi t$

At $t=0\text{sec}$, particle velocity is
$v_P=10\pi \times \cos 0^{\circ }=10\pi \text{m/s}$

At $t=3\text{sec}$
$v_P=10\pi \cos \left(3\pi \right)=-10\pi \text{m/s}$
similarly, at $t=0.5\text{sec}$
$v_P=10\pi \cos \left(\pi /2\right)=0\text{m/s}$