Question

A sinusoidal wave propagates along a string in figure (a) and (b). 'y' represents displacement of particle from the mean position. 'x' & 't' have usual meanings. Find:

1. Wavelength, frequency and speed of the wave

2. Maximum velocity and maximum acceleration of the particles

• A. $2m,0.25Hz,2m{s}^{-1},1.5mm{s}^{-1},0.75mm{s}^{-2}$
• B. $4m,0.25Hz,10m{s}^{-1},1.5\pi mm{s}^{-1},0.75mm{s}^{-2}$
• C. $4m,0.25Hz,1m{s}^{-1},1.5\pi mm{s}^{-1},0.75{\pi }^{2}mm{s}^{-2}$
• D. None of these

Answer 1

The correct option is C

$4m,0.25Hz,1m{s}^{-1},1.5\pi mm{s}^{-1},0.75{\pi }^{2}mm{s}^{-2}$

From the y-t graph we can see the wave repeats itself on interval of 4s. So the time period is 4s. from the y-x graph we see that the distance between the points after which the wave repeats itself is 4m. Hence the wave length is 4m.

$f=\frac{1}{T}=\frac{1}{4}{S}^{-1}=0.25s^{-1}$

$v=\lambda f=4\frac{1}{4}=1m{s}^{-1}$

$v_{p}max=A\omega$

$=A\times (2\pi f$

$=3\times \times 2\pi \times 0.25$ [A=3 from y vst graph]

$a_{p}max={\omega }^{2}A$

$=(2\pi \times 0.25{)}^2\times 3$

$\frac{3}{4}{\pi }^2=0.075{\pi }^{2}mm{s}^{-2}$