Question

a traveling disturbance on a string is represented by the equation y =0.2 sin (kx - At), where t is time in second. if the wave speed is teice the maximum particle speed then the value of k will be​

Answer 1

The disturbance is represented by the equation,

$\sf{\longrightarrow y=0.2\sin(kx-\omega t)}$

The particle speed is given by the first derivative of this equation wrt t, i.e.,

$\sf{\longrightarrow v_p=\dfrac{dy}{dt}}}$

$\sf{\longrightarrow v_p=\dfrac{d}{dt}\left[0.2\sin(kx-\omega t)\right]}$

$\sf{\longrightarrow v_p=0.2\omega\cos(kx-\omega t)}$

Maximum particle speed is obtained when $\sf{\cos(kx-\omega t)=1,}$ i.e.,

$\sf{\longrightarrow v_{p(max)}=0.2\omega$

The wave speed is given by,

$\sf{\longrightarrow v_w=\dfrac{\omega}{k}}$

Given that the wave speed is twice the maximum particle speed, i.e.,

$\sf{\longrightarrow v_w=2v_{p(max)}}$

$\sf{\longrightarrow\dfrac{\omega}{k}=0.4\omega}$

Since $\sf{\omega\neq0,}$

$\sf{\longrightarrow\dfrac{1}{k}=0.4}$

$\sf{\longrightarrow k=\dfrac{1}{0.4}}$

$\sf{\longrightarrow\underline{\underline{k=2.5}}}$

Answer 2

Answer:

THE Distribution is represented by the equation

➜ y = 0.2 sin ( kx - wt)

THE particles speed is given by the first derivate of this equation wrt ; that is;

$vp_{p} \: = \frac{dy}{dt}$

$vp_{p} = \frac{d}{dt} \\ \\ \\ vp = 0.2 \cos \: (kx - wt)$

max. particle speed obtained when

$\cos(kx - wt) = 1$

THE WAVE speed is given

$v_{w} \: = \frac{w}{k}$

GIVEN the wave speed is twice the max. particle speed ; i.e.;

$\\ v_{w} = 2vp(max) \\ \\ \frac{w}{k} = 0.4w$

since , w is not equal to 0

$\\ \\ \frac{1}{k} = 0.4 \\ \\ k = \frac{1}{0.4} \\ \\ k = 2.5 \\ \\ ans$