Question

A wave is described by the equation $y=\left(1.0mm\right)sin\pi (\frac{x}{2.0cm}-\frac{t}{0.01s})$
(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

$Y=\left(1mm\right)sin\pi [\frac{x}{2}cm-\frac{t}{0.01}sec]$

(a) $T=2\times 0.01=0.02sec=20ms$

$\lambda =2\times 2=4cm$

(b) $v=\frac{dy}{dt}=\frac{d}{dt}\left[sin2\pi \right(\frac{x}{4}-\frac{t}{0.02}\left)\right]$

$=-cos2\pi \left\{\right(\frac{x}{4})-(\frac{t}{0.02}\left)\right\}\times \frac{1}{0.02}$

$\Rightarrow v=-50cos2\pi \left\{\right(\frac{x}{4})-(\frac{t}{0.02}\left)\right\}atx=1andt=0.01sec,$

$v=-50cos2\pi \{\frac{1}{4}-\frac{1}{2}\}$

$=0$

(c) (i) at x = 3 cm, t = 0.01 sec,

$v=-50cos2\pi \{\frac{3}{4}-\frac{1}{2}\}=0$

(ii) at x = 5 cm, t = 0.01 sec,

v = 0 (putting the values)

at x = 7 cm, t = 0.1 sec,

v = 0
at x = 1 cm, t = 0.011 sec,

$v=50cos2\pi \left\{\right(\frac{1}{4})-(\frac{0.011}{0.02}\left)\right\}$

$=-50cos\left(\frac{3\pi }{5}\right)=-9.7cm/sec$

(similarly the other two can be calculated).