Question

A wave is described by the equation
y=(1·0 mm) sin π(x2·0 cm-t0·01 s).
(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⋅0 cm at time t = 0⋅01 s. (c) What are the speeds of the particles at x = 3⋅0 cm, 5⋅0 cm and 7⋅0 cm at t = 0⋅01 s?
(d) What are the speeds of the particles at x = 1⋅0 cm at t = 0⋅011, 0⋅012, and 0⋅013 s?

Answer 1

Wavelength is Ratio of Speed and Frequency

Explanation:

a) For A = 1.0 mm, $\lambda$ = 4.0 cm, T = 0.02 s

y = $1.0 sin\pi (\frac{x}{2.0} - \frac {t}{0.01})$

 = $1.0 sin2\pi (\frac{x}{4.0} - \frac {t}{0.02})$

 = $A sin2\pi (\frac{x}{\lambda} - \frac {t}{T})$

b) $\frac {\partial y}{\partial t} = -\omega A cos2\pi(\frac{x}{\lambda} - \frac {t}{T})$

= -$\frac {2\pi A}{T} cos2\pi(\frac{x}{\lambda} - \frac {t}{T})$

= $- \frac {2\pi \times 1.0 mm}{0.02 s} cos2\pi(\frac{x}{4.0} - \frac {t}{0.02 s})$

= $- \frac {\pi}{10}m/s\ cos2\pi(\frac{x}{2.0 cm} - \frac {t}{0.01 s})$

c) $v_p = (3.0,0.01) = - \frac {\pi}{10}\ cos\pi(\frac{3}{2} - 1)= 0\ m/s$

$v_p = (5.0,0.01 s) = - \frac {\pi}{10}ms\ cos\pi(\frac{5}{2} - 1)= 0\ m/s$

$v_p = (7.0,0.01) = - \frac {\pi}{10}m/s\ cos\pi(\frac{7}{2} - 1)= 0\ m/s$

d) $v_p(1.0 cm,0.011s) = -\frac {\pi}{10}cos \pi (\frac {1}{12} - 1.1)$

=  $-\frac {\pi}{10}cos 0.6\pi = -\frac {\pi}{10}cos \frac {3\pi}{5} = 9.7cm/s$

$v_p(1.0 cm,0.012s) = -\frac {\pi}{10}m/s\cos (\frac {1}{2} - \frac {0.012}{0.01})$

= $-\frac {\pi}{10}cos\pi (0.5 - 1.2)$

=  $-\frac {\pi}{10}cos0.7 \pi = 18.5 cm/s$

$v_p(1.0 cm,0.012s) = -\frac {\pi}{10}m/s$

$cos\pi (\frac {1}{2} - \frac {0.013}{0.01})$

= $-\frac {\pi}{10}cos 0.8\pi$

= $25.4 cm/s$

Answer 2

Given that,

A wave equation is

$y=1.0\sin\pi(\dfrac{x}{2.0}-\dfrac{t}{0.01})$...(I)

(a). We need to calculate the time period  and wavelength

Using wave equation

$y=a\sin2\pi(\dfrac{x}{\lambda}-\dfrac{t}{T})$

On comparing given wave equation

$y=1.0\sin2\pi(\dfrac{x}{4.0}-\dfrac{t}{0.02})$

The time period is 0.02 sec.

The wavelength is 4.0 cm.

(b). Given that,

x = 1.0 cm

time t = 0.01 s

We need to calculate the speed of the particle

Using formula of velocity

$v=\dfrac{dy}{dt}$

Put the value of y from wave equation

$v=-\omega A\cos2\pi(\dfrac{x}{\lambda}-\dfrac{t}{T}))$

$v=\dfrac{-2\pi A}{T}\cos2\pi(\dfrac{x}{\lambda}-\dfrac{t}{T}))$

Put the value into the formula

$v=\dfrac{2\pi\times1.0}{0.02}\cos2\pi(\dfrac{x}{4.0}-\dfrac{t}{0.02})$

$v=-\dfrac{\pi}{10}\cos\pi(\dfrac{x}{2.0}-\dfrac{t}{0.01})$....(II)

Put the value of x and t

$v=-\dfrac{\pi}{10}\cos\pi(\dfrac{1.0}{2.0}-\dfrac{0.01}{0.01})$

$v=0$

The speed of the particle is zero.

(c). We need to calculate the speed of the particle

(I). If x = 3.0 cm, t = 0.01 s

Put the value in the equation (II)

$v=-\dfrac{\pi}{10}\cos\pi(\dfrac{3.0}{2.0}-\dfrac{0.01}{0.01})$

$v=0$

(II). If x = 5.0 cm, t = 0.01 s

Put the value in the equation (II)

$v=-\dfrac{\pi}{10}\cos\pi(\dfrac{5.0}{2.0}-\dfrac{0.01}{0.01})$

$v=0$

(III). If x = 7.0 cm, t = 0.01 s

Put the value in the equation (II)

$v=-\dfrac{\pi}{10}\cos\pi(\dfrac{7.0}{2.0}-\dfrac{0.01}{0.01})$

$v=0$

(d). We need to calculate the speed of the particles

(I). at x = 1.0 , t = 0.011

Put the value in the equation (II)

$v=-\dfrac{\pi}{10}\cos\pi(\dfrac{1.0}{2.0}-\dfrac{0.011}{0.01})$

$v=9.7\ cm/s$

(II). at x = 1.0 , t = 0.012

Put the value in the equation (II)

$v=-\dfrac{\pi}{10}\cos\pi(\dfrac{1.0}{2.0}-\dfrac{0.012}{0.01})$

$v=18.5\ cm/s$  

(III). at x = 1.0 , t = 0.013

Put the value in the equation (II)

$v=-\dfrac{\pi}{10}\cos\pi(\dfrac{1.0}{2.0}-\dfrac{0.013}{0.01})$

$v=25.4\ cm/s$

Hence, This is the required answer.