Question

Repeat Problem 11:

(a) If$\mathit{q}\mathbf{=}\mathit{R}\mathit{e}\mathbf{4}{\mathit{e}}^{\mathbf{30}\mathbf{i}\mathbf{\pi }\mathbf{t}}$

(b) If$\mathit{q}\mathbf{=}\mathbf{lm}\mathbf{4}{\mathit{e}}^{\mathbf{30}\mathit{i}\mathit{\pi }\mathit{t}}$

Short answer

a)

1. The velocity amplitude is $120\pi i$.
2. Amplitude = 4
3. $Period=\frac{1}{15}$
4. Frequency = 15

b)

1. The velocity amplitude is $120\pi i$.
2. Amplitude = 4
3. role="math" localid="1659244633384" $Period=\frac{1}{15}$
4. Frequency = 15

Step-by-step solution

Given data

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

Concept of Periodic motion formula

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

Calculation of the real function q=Re4e30iπt 

Consider a particle whose coordinate is Re(z).

Now, $Re\left(q\right)=q=4\cos 30i\pi t$.

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

role="math" localid="1659245878662" $A\sin \omega t$[Or$A\cos \omega t$ or$A\sin \left(\omega t-\varphi \right)$ or $A\cos \left(\omega t-\varphi \right)$].

Hence, this particle is undergoing simple harmonic motion.

Now, $Re\left(q\right)=q=4\cos 30i\pi t$.

Amplitude = 4

$\omega =30i$

$$\begin{gathered} Period=\frac{2\pi }{\omega } \\ Period=\frac{2\pi }{30\pi } \\ Period=\frac{1}{15} \end{gathered}$$

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

Velocity amplitude$=A\omega$

Velocity amplitude $=120\pi i$

Calculation of the imaginary function q=lm4e30iττt

Consider a particle whose coordinate is lm(z).

Now, $lmq=q=4\sin 30i\pi t$.

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

$A\sin \omega t$[Or$A\cos \omega t$or$A\sin \left(\omega t-\varphi \right)$or$A\cos \left(\omega t-\varphi \right)$]

Hence, this particle is undergoing simple harmonic motion.

Now, $\mathrm{lm}q=q=4\sin 30i\pi t$.

Amplitude = 4

$\omega =30\pi i$

$\mathrm{Period}=\frac{2\mathrm{\pi }}{\mathrm{\omega }}$

$\mathrm{Period}=\frac{2\pi }{30\pi i}$

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

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

$Frequency=15i$

Velocity amplitude $=A\omega$

Velocity amplitude $=120\pi i$.