Question

A 100 Hz sinusoidal wave is travelling in the positive x-direction along a string with a linear mass density of $3.5\times {10}^{-3}kg/m$ and a tension of 35 N. At time t = 0, the point x = 0, has maximum displacement in the positive y direction. Next when this point has zero displacement the slope of the string is $\pi /20$. which of the following expression represent (s) the displacement of string as a function of x (in metre) and t (in second).

• A. $y=0.025cos(200\pi t-2\pi x)$
• B. $y=0.5cos(200\pi t-2\pi x)$
• C. $y=0.025cos(100\pi t-10\pi x)$
• D. $y=0.5cos(100\pi t-10\pi x)$

Answer 1

The correct option is A $y=0.025cos(200\pi t-2\pi x)$

Let the wave have the form $y=Asin(\omega t-kx+\varphi )$

Since the frequency is $100Hz$, $\omega =2\pi \nu =200\pi$

Speed of the wave=$\sqrt{\frac{T}{\mu }}=\frac{\omega }{k}$

$⟹k=2\pi$

Since displacement is maximum at (x,t)=(0,0), $sin(0+0+\varphi )=1$

$⟹\varphi =\frac{\pi }{2}$

Thus the wave is $y=Acos(\omega t-kx)$

$Slope=\left|\frac{dy}{dx}\right|=Aksin(\omega t-kx)=\frac{\pi }{20}$ at $(x,t)=(0,0)$

Thus $Ak=\frac{\pi }{20}$

$⟹A=0.025m$

Thus the correct answer is option A.