91Ó°ÊÓ

The position function is $\mathrm{x}=6\cos \left(3\mathrm{Ï€³Ù}+\frac{\mathrm{Ï€}}{3}\right)$.

The velocity function can be found by differentiating the position equation with respect to time and acceleration, and the acceleration is a derivative of the velocity with respect to time. To find the frequency, time period, and phase, we can compare the given equation of position with the standard equation.

Formulae:

The time period of a body in motion

$\mathrm{T}=\frac{1}{\mathrm{f}}$ (i)

The velocity of a body

$\mathrm{v}=\frac{d\mathrm{x}}{d\mathrm{t}}$ (ii)

The acceleration of a body

$\mathrm{a}=\frac{d\mathrm{v}}{d\mathrm{t}}$ (iii)

Angular frequency of a body in oscillation

$\mathrm{Ó\neg }=2×\mathrm{Ï€}×\mathrm{f}$ (iv)

Using the given equation of displacement of motion at$\mathrm{t}=2\sec$, we get the displacement of the motion as

$\begin{array}{rcl}\mathrm{x}& =& 6\cos \left(\left(3\mathrm{π}×2\right)+\frac{\mathrm{π}}{3}\right)\\ & =& 3\mathrm{m}\end{array}$

Hence, the value of displacement is 3 m.

Using equation (ii), we get the velocity by differentiating the given displacement equation with respect to time.

$$\begin{gathered} \mathrm{v}=\frac{\mathrm{d}}{d\mathrm{t}}\left(6\cos \left(3\mathrm{Ï€³Ù}+\frac{\mathrm{Ï€}}{3}\right)\right) \\ =-6×3\mathrm{Ï€²õ¾\pm ²Ô}\left(3\mathrm{Ï€³Ù}+\frac{\mathrm{Ï€}}{3}\right)................\left(\mathrm{a}\right)\left(âˆ\mu \mathrm{t}=2\sec \right) \\ =-49\mathrm{m}/\mathrm{s} \end{gathered}$$

Hence, the value of velocity is -49 m/s.

Using equation (ii) and from equation (a), we get the velocity by differentiating the given displacement equation with respect to time.

$\begin{array}{rcl}\mathrm{a}& =& \frac{\mathrm{d}}{d\mathrm{t}}\left(-6×3\mathrm{Ï€²õ¾\pm ²Ô}\left(3\mathrm{Ï€³Ù}+\frac{\mathrm{Ï€}}{3}\right)\right)\\ & =& -6×3\mathrm{Ï€}×3\mathrm{Ï€³¦´Ç²õ}\left(3\mathrm{Ï€³Ù}+\frac{\mathrm{Ï€}}{3}\right)\\ & =& -270\mathrm{m}/{\mathrm{s}}^2\end{array}$

Hence, the value of acceleration is $-270\mathrm{m}/{\mathrm{s}}^2$.

By comparing withthedisplacement equation of simple harmonic motion, the phase is calculated as
$\begin{array}{rcl}\mathrm{ϕ}& =& 3\mathrm{π}\left(2\right)+\frac{\mathrm{π}}{3}\\ & =& 19.987\mathrm{rad}\\ & ≈& 19.9\mathrm{rad}\\ & ≈& 20\mathrm{rad}\end{array}$

Hence, the phase value is 20 rad.

From equation of displacement,$\mathrm{Ó\neg }=3\mathrm{Ï€}\mathrm{rad}/\sec$.

Hence, using equation (iv), the frequency of the motion is calculated as

$\begin{array}{rcl}\mathrm{f}& =& \frac{\mathrm{Ó\neg }}{2\mathrm{Ï€}}\\ & =& 3\mathrm{Ï€}/2\mathrm{Ï€}\\ & =& 1.5\mathrm{Hz}\end{array}$

Hence, the value of frequency is 1.5 Hz.

Using equation (i), we get the period of motion as

$\begin{array}{rcl}\mathrm{T}& =& \frac{1}{1.5}\\ & =& 0.67\sec \end{array}$

Hence, the value of period is 0.67 sec.