91Ó°ÊÓ

The given complex function is z = f(t) and $z=-4e^{i\left(2t+3\mathrm{Ï€}\right)}$.

Time Period (T): The length of time it takes for a motion to repeat itself. Thus, a time period is measured in seconds.

Frequency (f): It is determined by counting how many times a motion is repeated in a second. Hz is the symbol for frequency (Hertz).

Frequency is related to Time period as $\mathit{f}\mathbf{=}\frac{\mathbf{1}}{\mathbf{T}}$.

The function given below is the complex function.

$\begin{array}{l}Z=f\left(t\right)\\ Z=-4cos(2t+3\mathrm{Ï€})+\mathrm{i}sin(2t+3\mathrm{Ï€})\end{array}$

Consider a particle whose coordinate is Re(z).

Now, $Re\left(z\right)=4\cos \left(2t+3\mathrm{Ï€}\right)$.

By definition an object is executing simple harmonic motion if its displacement from equilibrium can be written as $A\sin \mathrm{Ó\neg }t$[Or $A\cos \mathrm{Ó\neg }t$or $A\sin \left(\mathrm{Ó\neg }t-\mathrm{Ï•}\right)$or $A\cos \left(\mathrm{Ó\neg }t-\mathrm{Ï•}\right)$]

Hence, this particle is undergoing simple harmonic motion.

Now, for coordinate, $Re\left(z\right)=-4\cos (2t+3\mathrm{Ï€})$.

Amplitude = -4

$\mathrm{Ó\neg }=2$

For period:

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

Period $=\mathrm{Ï€}$

For frequency:

Frequency $=\frac{1}{\mathrm{period}}$

Frequency $=\frac{1}{\mathrm{Ï€}}$

Velocity amplitude $=A\mathrm{Ó\neg }$

Velocity amplitude = -8

Following is the complex function:

$\begin{array}{l}Z=f\left(t\right)\\ Z=-4cos(2t+3\mathrm{Ï€})+\mathrm{i}sin(2t+3\mathrm{Ï€})\end{array}$

Consider a particle whose coordinate is lm z .

Now, $\mathrm{lm}z=-4\sin \left(2t+3\mathrm{Ï€}\right)$.

By definition an object is executing simple harmonic motion if its displacement from equilibrium can be written as:

$A\sin \mathrm{Ó\neg }t$[Or$A\cos \mathrm{Ó\neg }t$ or $A\sin \left(\mathrm{Ó\neg }t-\mathrm{Ï•}\right)$or$A\cos \left(\mathrm{Ó\neg }t-\mathrm{Ï•}\right)$]

Hence, this particle is undergoing simple harmonic motion

Now, for coordinate, $\mathrm{lm}z=-4\sin \left(2t+3\mathrm{Ï€}\right)$.

Amplitude = 4

$\mathrm{Ó\neg }=2$

For period:

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

Period$=\mathrm{Ï€}$

For frequency:

Frequency$=\frac{1}{\mathrm{period}}$

Frequency$=\frac{1}{\mathrm{Ï€}}$

Velocity amplitude $=A\mathrm{Ó\neg }$

Velocity amplitude = -8