Find the amplitude, period, frequency, and velocity amplitude for the motion of a particle whose distance from the origin is the given function.

$\mathit{s}\mathbf{=}\mathbf{3}\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathbf{2}\mathit{t}\mathbf{+}\mathit{\pi }\mathbf{/}\mathbf{8}\mathbf{)}\mathbf{+}\mathbf{3}\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathbf{2}\mathit{t}\mathbf{-}\mathit{\pi }\mathbf{/}\mathbf{8}\mathbf{)}$

Distance from the origin is the given function .$s=3\sin (2t+\pi /8)+3\sin (2t-\pi /8)$

Time Period (T): It is the time taken by the motion to repeat itself. So the unit of a time period is seconds.

Frequency (f) : It is defined as a number of times the motion is repeated in one second. The unit of frequency is Hz (Hertz).

Frequency is related to Time period as f = 1/T .

Following is the distance function:

$\begin{array}{rcl}s& =& 3\sin \left(2t+\frac{\pi }{8}\right)+3\sin \left(2t-\frac{\pi }{8}\right)\\ s& =& 3\left(\sin \left(2t+\frac{\pi }{8}\right)+\sin \left(2t-\frac{\pi }{8}\right)\right)\\ s& =& 6\sin 2t\cos \frac{\pi }{8}\\ & =& \left(6\cos \frac{\pi }{8}\right)\sin 2t\end{array}$

Amplitude is the maximum value of distance from the mean position.

So, amplitude is $6\cos \frac{\pi }{8}$.

The distance functions of the form.$A\sin \omega t$

So,.$\omega =2$

Then, obtain values:

Period $=\frac{2\pi }{\omega }$

Period $=\frac{2\pi }{2}$

Period $=\pi$

The frequency is defined as the inverse of the period.

So,.$\text{Frequency}=\frac{1}{\text{period}}$

Frequency$=\frac{1}{\pi }$

$$\begin{gathered} s=\left(6\cos \frac{\pi }{8}\right)\sin 2t \\ \frac{ds}{dt}=\left(12\cos \frac{\pi }{8}\right)\cos 2t \end{gathered}$$

So, velocity amplitude $=12\cos \frac{\pi }{8}$.