Question

A wave is described by the equation

$y=1.0mm\mathrm{sin\pi }\left(\frac{x}{2.0cm}-\frac{t}{0.01s}\right)$

(a) Find the time period and the wavelength?

(b) Write the equation for the velocity of the particles. Find the speed of the particle at $x=1.0cm$ at time $t=0.01s$

(c) What are the speeds of the particles at $x=3.0cm$, $5.0cm$ and $7.0cm$ at $t=0.01s$?

Answer 1

$\left(a\right)$ Step 1.Given data

Equation of wave $y=1.0mm\sin \mathrm{\pi }\left(\frac{\mathrm{x}}{2.0\mathrm{cm}}-\frac{\mathrm{t}}{0.01\mathrm{s}}\right)$ ………… $eq\left(1\right)$

Step 2. Comparing the given equation to the general equation of the wave

General equation of the wave

$y=A\sin \left(kx-\omega t\right)$

where $A$ is amplitude of the wave

$k$ is the wave vector

$\omega$ is the angular frequency

Now, comparing the given equation to the general equation of the wave ,we get

$A=1mm$

$k=\frac{\mathrm{\pi }}{2}c{m}^{-1}$

$\omega =100rad{s}^{-1}$

Step 3. Finding the time period of the wave

By using the formula of the time period

$T=\frac{2\mathrm{\pi }}{\omega }$

$T=\frac{2\mathrm{\pi }}{100\mathrm{\pi }}$

$T=0.02s$

Step 4. Finding the wavelength of the wave

By using the formula of the wavelength

$\lambda =\frac{2\mathrm{\pi }}{k}$

$\lambda =\frac{2\mathrm{\pi }}{\mathrm{\pi }}\times 2$

$\lambda =4cm$

Hence , the time period of the wave is$0.02s$ and wavelength of the wave is$4cm$

$\left(b\right)$ Step 1. Given data

$x=1.0cm$

$t=0.01s$

Step 2. Finding the velocity of wave

If we derivate equation $1$ with respect to time then, we get

velocity ,$v=\frac{dy}{dt}$

$v=-\omega A\cos \left(kx-\omega t\right)$ ………..$eq\left(2\right)$

Step 3. Finding the velocity at $x=1.0cm$ and $t=0.01s$

Now putting the given values in $eq\left(2\right)$, we get

$v=-100\mathrm{\pi }\cos \left(\frac{\mathrm{\pi x}}{2}-100\mathrm{\pi t}\right)$

$v=-100\mathrm{\pi }\cos \left(\frac{\mathrm{\pi }}{2}-\mathrm{\pi }\right)$

$v=0$

Hence the velocity at $x=1.0cm$is $0$

$\left(c\right)$ Step 1. Given data

$x=3cm,5cm,7cm$

Step 2. Finding the velocity at $x=3cm$

Now putting $x=3cm$ in $eq\left(2\right)$ ,we get

$v=-100\mathrm{\pi }\cos \left(\frac{3\mathrm{\pi }}{2}-\mathrm{\pi }\right)$

$v=0$

Step 3. Finding the velocity at $x=5cm$

Putting $x=5cm$ in $eq\left(2\right)$, we get

$v=-100\mathrm{\pi }\cos \left(\frac{5\mathrm{\pi }}{2}-\mathrm{\pi }\right)$

$v=0$

Step 4. Finding the velocity at $x=7cm$

Putting $x=7cm$ in $eq\left(2\right)$, we get

$v=-100\mathrm{\pi }\cos \left(\frac{7\mathrm{\pi }}{2}-\mathrm{\pi }\right)$

$v=0$

Hence , velocity at $x=3cm,5cm,7cm$ is $0$.