Q2P - Mathematical Methods in Physical Sciences

Question

In each of the following problems, z represents the displacement of a particle from the origin. Find (as functions of t) its speed and the magnitude of its acceleration, and describe the motion.

$\mathbf{z}\mathbf{=}\mathbf{-}\mathbf{5}{\mathbf{e}}^{\mathbf{i\omega t}}\mathbf{,}\mathbf{\omega }\mathbf{=}\mathbf{const}$

Step-by-Step Solution

Answer

 The part it follows is circular, $x^2+y^2=25$.

Its speed is $5\omega$.

Its acceleration is $5{\omega }^2$.

Step 1: Given Information.

The given expression is $\mathrm{z}=-5{\mathrm{e}}^{\mathrm{i\omega t}}$.

Step 2: Meaning of rectangular form.

Represent the complex number in rectangular form means writing the given complex number in the form of x + iy in which x is the real part and y is the imaginary part.

Step 3: Find the magnitude.

Write the general form of the complex number.

$$\begin{gathered} \mathrm{z}=\mathrm{x}+yi \\ \mathrm{z}=a{\mathrm{e}}^{\left(\mathrm{i\omega t}\right)} \end{gathered}$$

Find the magnitude.

$$\begin{gathered} \left|z\right|=\sqrt{x^2+y^2} \\ \left|z\right|=5 \end{gathered}$$

Hence $x^2+y^2=25$.

Step 4: Find the velocity and acceleration.

Find the velocity.

$$\begin{gathered} v=a\omega \\ v=5\omega \end{gathered}$$

Find the acceleration.

$$\begin{gathered} A=a{\omega }^2 \\ A=5{\omega }^2 \end{gathered}$$

Therefore, the path it follows it circular $x^2+y^2=25$.

Its speed is $5\omega$.

Its acceleration is $5{\omega }^2$.