The equation of a progressive wave travelling along a string is given by $$y = 10 \,sin\pi \,(0.01x - 2.00t)$$ where x and y are in centimetres and t in seconds. Find the (i) velocity of a particle at $$x = 2 \,m$$ and $$t = \dfrac{5}{6 }\,s.$$ (ii) acceleration of a particle at $$x = 1 \,m$$ and $$t = \dfrac{1}{4} \,s.$$ Also find the velocity amplitude and acceleration amplitude for the wave.
The displacement equation for a particle is given by
$$y = 10 sin \pi (0.01 x - 2.00 t)cm$$ (i)
i. Particle velocity is given by
$$\dfrac {dy}{dt} = -20.0\pi cos (0.01 x - 2.00 t) cm/s$$ (ii)
Putting $$x = 200 \,cm$$ and $$t = \dfrac{5}{6} \,s,$$ we get
$$\dfrac {dy}{dt} = -20.0 \pi cos \pi \left ( 2 - \dfrac {5}{3} \right )cm/s$$
Also, the velocity amplitude
$$\dfrac {dy}{dt}\left. \right \}_{max} = 20.0 \pi \,cm/s$$
ii. Differentiating Eq. (ii) w.r.t. time t, we get the particle acceleration as
$$\dfrac {d^2y}{dt^2} = -40.0 \pi^2 sin \pi (0.01 \pi - 2.00 t) cm/s^2$$
Putting $$x = 100 \,cm$$ and $$t = \dfrac{1}{4 }\,s,$$ we get
$$\dfrac {d^2y}{dt^2} = -40.0 \pi^2 sin \pi\left ( 1 - \dfrac {1}{2} \right )cm/s^2$$
$$ = -40.0 \pi^2 \,cm/s^2$$
Also, the acceleration amplitude
$$\dfrac {d^2y}{dt^2}\left. \right \}_{max} = 40.0 \pi^2 \,cm/s^2$$
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