The function $\mathbf{x}\mathbf{=}\left(6.0m\right)\mathbf{cos}\left[\left(3\mathrm{\pi rad}/s\right)t+\frac{\pi }{3}\mathrm{rad}\right]$ gives the simple harmonic motion of a body. At data-custom-editor="chemistry" $\mathbf{t}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{s}$,

1. What is the displacement of the motion?
2. What is the velocity of the motion?
3. What is the acceleration of the motion?
4. What is the phase of the motion?
5. What is the frequency of the motion?
6. What is the period of the motion?

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

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

$\mathrm{\omega }=2\times \mathrm{\pi }\times \mathrm{f}$ (iv)

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

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

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.

$$\begin{gathered} \mathrm{v}=\frac{\mathrm{d}}{d\mathrm{t}}\left(6\cos \left(3\mathrm{\pi t}+\frac{\mathrm{\pi }}{3}\right)\right) \\ =-6\times 3\mathrm{\pi sin}\left(3\mathrm{\pi t}+\frac{\mathrm{\pi }}{3}\right)................\left(\mathrm{a}\right)\left(\because \mathrm{t}=2\sec \right) \\ =-49\mathrm{m}/\mathrm{s} \end{gathered}$$

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.

$\begin{array}{rcl}\mathrm{a}& =& \frac{\mathrm{d}}{d\mathrm{t}}\left(-6\times 3\mathrm{\pi sin}\left(3\mathrm{\pi t}+\frac{\mathrm{\pi }}{3}\right)\right)\\ & =& -6\times 3\mathrm{\pi }\times 3\mathrm{\pi cos}\left(3\mathrm{\pi t}+\frac{\mathrm{\pi }}{3}\right)\\ & =& -270\mathrm{m}/{\mathrm{s}}^2\end{array}$

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
$\begin{array}{rcl}\mathrm{\varphi }& =& 3\mathrm{\pi }\left(2\right)+\frac{\mathrm{\pi }}{3}\\ & =& 19.987\mathrm{rad}\\ & \approx & 19.9\mathrm{rad}\\ & \approx & 20\mathrm{rad}\end{array}$

Hence, the phase value is 20 rad.

From equation of displacement,$\mathrm{\omega }=3\mathrm{\pi }\mathrm{rad}/\sec$.

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

$\begin{array}{rcl}\mathrm{f}& =& \frac{\mathrm{\omega }}{2\mathrm{\pi }}\\ & =& 3\mathrm{\pi }/2\mathrm{\pi }\\ & =& 1.5\mathrm{Hz}\end{array}$

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.