91Ó°ÊÓ

The given function is $q=\mathrm{Re}4e^{30i\mathrm{Ï€}t}$and $q=4\cos 30i\mathrm{Ï€}t$.

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

Consider a particle whose coordinate is Re(z).

Now, $Re\left(q\right)=q=4\cos 30i\mathrm{Ï€}t$.

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

role="math" localid="1659245878662" $A\sin \mathrm{Ó\neg }t$[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, $Re\left(q\right)=q=4\cos 30i\mathrm{Ï€}t$.

Amplitude = 4

$\mathrm{Ó\neg }=30i$

$$\begin{gathered} Period=\frac{2\mathrm{Ï€}}{\mathrm{Ó\neg }} \\ Period=\frac{2\mathrm{Ï€}}{30\mathrm{Ï€}} \\ Period=\frac{1}{15} \end{gathered}$$

$$\begin{gathered} Frequency=\frac{1}{Period} \\ Frequency=15 \end{gathered}$$

Velocity amplitude$=A\mathrm{Ó\neg }$

Velocity amplitude $=120\mathrm{Ï€}i$

Consider a particle whose coordinate is lm(z).

Now, $lmq=q=4\sin 30i\mathrm{Ï€}t$.

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, $\mathrm{lm}q=q=4\sin 30i\mathrm{Ï€}t$.

Amplitude = 4

$\mathrm{Ó\neg }=30\mathrm{Ï€}i$

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

$\mathrm{Period}=\frac{2\mathrm{Ï€}}{30\mathrm{Ï€}i}$

$\mathrm{Period}=\frac{1}{15i}$

$Frequency=\frac{1}{Period}$

$Frequency=15i$

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

Velocity amplitude $=120\mathrm{Ï€}i$.