A wave on a string is described by\(y\left( {x,t} \right) = Acos\left( {kx - \omega t} \right)\). (a) Graph \(y,\,{v_y}\,and\,{a_y}\)as functions of \(x\) for time\(t = 0\). (b) Consider the following points on the string:\(\left( i \right) x = 0;\left( {ii} \right) x = {\pi \mathord{\left/

{\vphantom {\pi {4K}}} \right.

\\} {4K}}; \left( {iii} \right) x = {\pi \mathord{\left/

{\vphantom {\pi {2K}}} \right.

\\} {2K}}; \left( {iv} \right) x = 3{\pi \mathord{\left/

{\vphantom {\pi {4K}}} \right.

\\} {4K}}; \left( v \right) x = {\pi \mathord{\left/

{\vphantom {\pi K}} \right.

\\} K};\)\(\left( {vi} \right) x = {{5\pi } \mathord{\left/

{\vphantom {{5\pi } {4k}}} \right.

\nulldelimiterspace} {4k}}; \left( {vii} \right)x = {{3\pi } \mathord{\left/

{\vphantom {{3\pi } {2k}}} \right.

\\} {2k}}; \left( {viii} \right)x = {{7\pi } \mathord{\left/

{\vphantom {{7\pi } {4k}}} \right.

\\} {4k}}\). For a particle at each of these points at\(t = 0\), describe in words whether the particle is moving and in what direction, and whether the particle is speeding up, slowing down, or instantaneously not accelerating.

Step by step solution

01

Given data

\(y\left( {x,t} \right) = Acos\left( {kx - \omega t} \right)\)

02

Concept/ Formula used

\(\begin{aligned}{l}{v_y} = \frac{{dy}}{{dt}}\\{a_y} = \frac{{{d^2}y}}{{d{t^2}}}\end{aligned}\)

03

Graph of \(y,\,{v_y}\,and\,{a_y}\)

(a)

\(\begin{aligned}{l}y\left( {x,t} \right) = Acos\left( {kx - \omega t} \right)\\{v_y} = A\omega \sin \left( {kx - \omega t} \right)\\{a_y} = - A{\omega ^2}\cos \left( {kx - \omega t} \right)\end{aligned}\)

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