Question

A simple harmonic wave of amplitude 1 cm and frequency 100 is travelling along positive X- direction with a velocity of 15m/s. Calculate the displacement y, particle velocity and x=180cm from the origin at t=5 s.

Answer 1

$$\begin{gathered} Given \\ A=1cm \\ f=100Hz \\ v=15m{s}^{-1} \\ Weknow \\ \omega =2\mathrm{\pi f}=2\mathrm{\pi }\times 100=200\mathrm{\pi } \\ \mathrm{v}=\frac{\mathrm{\omega }}{\mathrm{k}} \\ \mathrm{k}=\frac{\mathrm{\omega }}{\mathrm{v}}=\frac{200\mathrm{\pi }}{15}=\frac{40\mathrm{\pi }}{3} \\ \mathrm{If}\mathrm{wave}\mathrm{travels}\mathrm{in}\mathrm{positive}\mathrm{direction}\mathrm{then}\mathrm{its}\mathrm{equation}\mathrm{is}\mathrm{given}\mathrm{by} \\ \mathrm{y}=\mathrm{A}\sin \left(\mathrm{\omega t}-\mathrm{kx}\right) \\ \mathrm{y}=\sin \left(200\mathrm{\pi t}-\frac{40\mathrm{\pi }}{3}\mathrm{x}\right)\mathrm{cm} \\ \mathrm{partice}\mathrm{velocity},{\mathrm{v}}_{\mathrm{p}}=\frac{\partial \mathrm{y}}{\partial \mathrm{t}}=200\mathrm{\pi }\cos \left(200\mathrm{\pi t}-\frac{40\mathrm{\pi }}{3}\mathrm{x}\right)\mathrm{cm}{\mathrm{s}}^{-1} \\ \mathrm{At}\mathrm{t}=5\mathrm{s}\mathrm{and}\mathrm{x}=180\mathrm{cm}=1.8\mathrm{m} \\ \mathrm{y}=\sin \left(200\mathrm{\pi }\times 5-\frac{40\mathrm{\pi }}{3}\times 1.8\right)\mathrm{cm} \\ =\sin \left(2\mathrm{\pi }\times 500-40\mathrm{\pi }\times 0.6\right)\mathrm{cm} \\ =\sin \left(-40\mathrm{\pi }\times 0.6\right)\mathrm{cm} \\ ==\sin \left(-24\mathrm{\pi }\right)\mathrm{cm} \\ =0 \\ {\mathrm{v}}_{\mathrm{p}}=200\mathrm{\pi }\cos \left(200\mathrm{\pi }\times 5-\frac{40\mathrm{\pi }}{3}\times 1.8\right)\mathrm{cm}{\mathrm{s}}^{-1} \\ =200\mathrm{\pi }\cos \left(2\mathrm{\pi }\times 500-40\mathrm{\pi }\times 0.6\right)\mathrm{cm}{\mathrm{s}}^{-1} \\ =200\mathrm{\pi }\cos \left(-40\mathrm{\pi }\times 0.6\right)\mathrm{cm}{\mathrm{s}}^{-1} \\ =200\mathrm{\pi }\cos \left(-24\mathrm{\pi }\right)\mathrm{cm}{\mathrm{s}}^{-1} \\ =200\mathrm{\pi }\cos \left(24\mathrm{\pi }\right)\mathrm{cm}{\mathrm{s}}^{-1} \\ =200\mathrm{\pi }\cos \left(2\mathrm{\pi }\times 12\right)\mathrm{cm}{\mathrm{s}}^{-1} \\ =200\mathrm{\pi cm}{\mathrm{s}}^{-1} \end{gathered}$$