Question

In each case, show that a particle whose coordinate is (a) x = Re z , (b) y =Im z , is undergoing simple harmonic motion, and find the amplitude, period, frequency, and velocity amplitude of the motion.

$\mathit{z}\mathbf{=}\mathbf{5}{\mathit{e}}^{\mathbf{i}\mathbf{t}}$

Short answer

The velocity amplitude is =5 .

Period =$2\pi$

Frequency = $\frac{1}{2\pi }$

Amplitude = 5

Step-by-step solution

Given data

The given complex function is and $z=f\left(t\right)\mathrm{and}z=5e^{it}$.

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

Following is the complex function:

z = f ( t )

z = 5 (cos t + i sin t)

Consider a particle whose coordinate is Re (z).

Now, Re(z) = 5cos t.

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

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

Hence, this particle is undergoing simple harmonic motion.

Calculation for the velocity amplitude

Now, for coordinate, Re(z)= 5cos t .

Amplitude = 5

$\omega =1$

For period of a particle:

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

Period =$2\pi$

For frequency of a particle:

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

Frequency =$\frac{1}{2\pi }$

Velocity amplitude = A$\omega$

Velocity amplitude =5.

Calculation for the imaginary function

Following is the complex function:

z = f ( t )

z = 5 (cos t + i sin t)

Consider a particle whose coordinate is Im(z).

Now, Imz = 5sin t.

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

$A\sin \omega t\left[OrA\cos \omega torA\sin \left(\omega t-\varphi \right)oraA\cos \left(\omega t-\varphi \right)\right]$

Hence, this particle is undergoing simple harmonic motion.

Calculation for the velocity amplitude

Now, for coordinate, Imz = 5sin t .

Amplitude = 5

$\omega =1$

For period:

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

Period =$2\pi$

For frequency:

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

Frequency =$\frac{1}{2\pi }$

Velocity amplitude = A$\omega$

Velocity amplitude = 5 .