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

$\mathit{s}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathit{c}\mathit{o}\mathit{s}\mathbf{(}\mathit{\pi }\mathit{t}\mathbf{-}\mathbf{8}\mathbf{)}$

Distance from the origin is the given function:

$s=\frac{1}{2}\cos (\pi t-8)$

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$\mathit{f}\mathbf{=}\frac{\mathbf{1}}{\mathbf{T}}$ .

Following is the distance function$s=\frac{1}{2}\cos (\pi t-8)$

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

So, amplitude = 1/2 .

The distance functions of the form $A\cos (\omega t-\varphi )$.

So, $\omega =\pi$.

Then, obtain values:

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

Period = 2

The frequency is defined as the inverse of the period.

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

Frequency $=\frac{1}{2}$

$\begin{array}{rcl}s& =& \frac{1}{2}\cos (\pi t-8)\\ \frac{ds}{dt}& =& -\frac{\pi }{2}\sin (\pi t-8)\end{array}$

Then, velocity amplitude is $\frac{\pi }{2}$.