Question

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?

Short answer

1. Displacement of the motion is 3 m.
2. Velocity of the motion is -49 m/s.
3. Acceleration of the motion is $-270\mathrm{m}/{\mathrm{s}}^2$.
4. Phase of the motion is 20 rad.
5. Frequency of the motion is 1.5 Hz.
6. Period of the motion is 0.67 sec.

Step-by-step solution

Stating the given data

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

Understanding the concept of motion

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)

a) Calculation of displacement

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.

b) Calculation of velocity

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.

c) Calculation of acceleration

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 $-270\mathrm{m}/{\mathrm{s}}^2$.

d) Calculation of phase

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.

e) Calculation of frequency

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.

f) Calculation of period

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.