Question 63: Waves

Question

A transverse sinusoidal wave moves along a string in the positive x-direction at a speed of 10 cm/s. The wavelength of the waves is 0.5 m and its amplitude is 10 cm. At a particular time t, the snap-shot of the wave is shown in figure. The velocity of point P when its displacement is 5 cm is :

Accepted Answer

Velocity of point $$p=dy/dt$$
= $$-$$(velocity of wave $$	imes$$ $$dy/dx$$)
Here, $$dy/dx=$$ negative ($$-$$ve). Therefore, velocity at point P is positive and is along X-axis only.
Equation of a wave moving in positive X-axis is given as,
$$y = A\sin (wt - \phi )$$ .... (i)
or, $${V_P} = Aw\cos (wt - \phi )$$ .... (ii)
Here, $$y' = 5$$ cm, $$A = 10$$ cm
$$	herefore$$ $$5 = 10\sin (wt - \phi ) \Rightarrow wt - \phi  = 30^\circ $$ .... (iii)
Put equation (iii) in equation (ii),
$${V_P} = 0.10 	imes w\cos 30^\circ $$
Now, $$V = {V_\lambda }$$, $$	herefore$$ $$v = V/\lambda  = {{0.10} \over {0.5}} = 0.2$$
$$	herefore$$ $$w = 2\pi v = 2\pi  	imes 0.2 = 0.4\pi $$
$$ \Rightarrow {V_P} = 0.1 	imes 0.4\pi  	imes {{\sqrt 3 } \over 2} = {{\sqrt 3 } \over {50}}\pi \,\widehat j$$ m/s
It is in positive Y-direction.

Answer

$${{\sqrt {3\pi } } \over {50}}\widehat j$$ m/s

Answer

$$ - {{\sqrt {3\pi } } \over {50}}\widehat j$$ m/s

Answer

$${{\sqrt {3\pi } } \over {50}}\widehat i$$ m/s

Answer

$$ - {{\sqrt {3\pi } } \over {50}}\widehat i$$ m/s

Waves

A transverse sinusoidal wave moves along a string in the positive x-direction at a speed of 10 cm/s. The wavelength of the waves is 0.5 m and its amplitude is 10 cm. At a particular time t, the snap-shot of the wave is shown in figure. The velocity of point P when its displacement is 5 cm is :

Choose the correct answer: