91Ó°ÊÓ

Distance from the origin is the given function .$s=3\sin (2t+\mathrm{Ï€}/8)+3\sin (2t-\mathrm{Ï€}/8)$

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:

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

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

So, amplitude is $6\cos \frac{\mathrm{Ï€}}{8}$.

The distance functions of the form.$A\sin \mathrm{Ó\neg }t$

So,.$\mathrm{Ó\neg }=2$

Then, obtain values:

Period $=\frac{2\mathrm{Ï€}}{\mathrm{Ó\neg }}$

Period $=\frac{2\mathrm{Ï€}}{2}$

Period $=\mathrm{Ï€}$

The frequency is defined as the inverse of the period.

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

Frequency$=\frac{1}{\mathrm{Ï€}}$

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

So, velocity amplitude $=12\cos \frac{\mathrm{Ï€}}{8}$.