Q14E - University Physics with Modern Physics

Question

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

Answer

(a)

Step 1: Given data

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

Step 2: Concept/ Formula used

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

Step 3: Graph of \(y,\,{v_y}\,and\,{a_y}\)

(a)

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