91Ó°ÊÓ

The given complex function is$z=f\left(t\right)$and $z=2e^{-it/2}$.

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

Following is the complex function:

$\begin{array}{l}z=f\left(t\right)\\ z=2\left(\cos \frac{t}{2}+i\sin \frac{t}{2}\right)\end{array}$

Consider a particle whose coordinate is Re(z).

Now, $\mathrm{Re}\left(z\right)=2cos\frac{t}{2}$.

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

$A\sin \mathrm{Ó\neg }t$[Orrole="math" localid="1659240088045" $A\cos \mathrm{Ó\neg }t$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, $\mathrm{Re}\left(z\right)=2cos\frac{t}{2}$.

Amplitude = 2

$\mathrm{Ó\neg }=1/2$

For period:

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

For frequency:

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

Frequency $=\frac{1}{4\mathrm{Ï€}}$

Period $=4\mathrm{Ï€}$

Velocity amplitude$=A\mathrm{Ó\neg }$

Velocity amplitude = 1

Following is the complex function:

$\begin{array}{l}z=f\left(t\right)\\ z=2\left(\cos \frac{t}{2}+i\sin \frac{t}{2}\right)\end{array}$

Consider a particle whose coordinate is lm(z).

Now, $\mathrm{lm}z=2sin\frac{t}{2}$.

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, $\mathrm{lm}z=2sin\frac{t}{2}$.

Amplitude = 2

$\mathrm{Ó\neg }=1/2$

For period:

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

Period$=4\mathrm{Ï€}$

For frequency:

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

Frequency $=\frac{1}{4\mathrm{Ï€}}$

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

Velocity amplitude = 1