Chapter 7: Problem 10

You are given a complex function $z=f(t) .$ In each case, show that a particle whose coordinate is (a) $x=\operatorname{Re} z,$ (b) $y=\operatorname{Im} z$ is undergoing simple harmonic motion, and find the amplitude, period, frequency, and velocity amplitude of the motion. $$\quad z=-4 e^{i(2 t+3 \pi)}$$

Short Answer

Expert verified

Both x and y are in simple harmonic motion with amplitude 4, period $\pi$, frequency $\frac{1}{\pi}$, and velocity amplitude 8.

Step by step solution

Express the complex function in terms of its real and imaginary parts

Given the complex function is z = -4 e^{i(2t + 3\pi)} . We rewrite $z$ using Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$. z = -4 (\cos(2t + 3\pi) + i\sin(2t + 3\pi)) Simplifying with trigonometric identities: $\cos(2t + 3\pi) = \cos 2t \text{, because } \cos(\theta + \pi) = -\cos \theta$. Therefore, real and imaginary parts become: x = 4 \cos(2t) and y = 4 \sin(2t)

Show that x is undergoing simple harmonic motion

Express the coordinate $x = 4\cos(2t)$. This is in the form of $A\cos(\omega t)$, which indicates simple harmonic motion where amplitude $A = 4$ and angular frequency $\omega = 2$.

Show that y is undergoing simple harmonic motion

Express the coordinate $y = 4\sin(2t)$. This is in the form of $A\sin(\omega t)$, which indicates simple harmonic motion where amplitude $A = 4$ and angular frequency $\omega = 2$.

Calculate the amplitude

Amplitude $A$ for both $x$ and $y$ is given directly as $4$.

Calculate the period

The period $T$ is determined by the angular frequency $\omega$. For $\omega = 2$, we use the formula $T = \frac{2\pi}{\omega}$ to find $T = \frac{2\pi}{2} = \pi$.

Calculate the frequency

The frequency $f$ is the reciprocal of the period: $f = \frac{1}{T} = \frac{1}{\pi}$.

Calculate the velocity amplitude

For simple harmonic motion, velocity amplitude is computed as $A\omega$. Given $A = 4$ and $\omega = 2$, the velocity amplitude is $A\omega = 4 \times 2 = 8$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Function

A **complex function** involves both real and imaginary parts. In this exercise, the complex function given is $ z = -4 e^{i(2t + 3\pi)} $. Essentially, complex functions help describe oscillating quantities like those found in harmonic motion. The function combines both cosine and sine elements, which makes it perfect for describing periodic behaviors. Rewriting it using Euler's formula simplifies our analysis.

Euler's Formula

For breaking down complex functions, **Euler’s Formula** is invaluable. It states that \[ e^{i\theta} = \cos \theta + i \sin \theta \]. By applying Euler’s Formula to the complex function $ z = -4 e^{i(2t + 3\pi)} $, we can rewrite it as: $-4 (\cos (2t + 3\pi) + i \sin(2t + 3\pi)) $. This representation separates the function into its real part, $x = 4 \cos 2t $, and its imaginary part, $y = 4 \sin 2t$, laying the groundwork for recognizing simple harmonic motion (SHM).

Angular Frequency

In the equations $ x = 4 \cos(2t) $ and $ y = 4 \sin(2t) $, the term 2 in the arguments of the cosine and sine functions is the **angular frequency** ($ \omega $). It tells us how fast the particle is oscillating. The formula for angular frequency is defined as: $ \omega = 2\pi f $, where $ f $ is the frequency. In our case, angular frequency $ \omega $ is 2 radians per second, indicating the oscillation rate of the particle.

Amplitude

The **amplitude** ($ A $) of the harmonic motion refers to the maximum extent of displacement from the equilibrium position. In the cases $ x = 4 \cos (2t) $ and $ y = 4 \sin (2t) $, the amplitude is clearly 4. Whether considering the real part, $ x $, or the imaginary part, $ y $, the maximum displacement from the mean position is 4 units. Amplitude is an important feature because it measures the energy of the oscillation.

Period

The **period** ($ T $) is the time it takes for one complete cycle of oscillation. It is inversely related to the frequency. For an angular frequency $ \omega = 2 $, we calculate the period using the formula: $ T = \frac{2\pi}{\omega} $. Hence, $ T = \frac{2\pi}{2} = \pi \, \text {seconds} $. Thus, it takes $ \pi $ seconds for the particle to complete one full oscillation cycle.

Frequency

The **frequency** ($ f $) of a harmonic motion indicates how many oscillations occur in one second. It is the reciprocal of the period. So, with our period $ T = \pi $, the frequency is $ f = \frac{1}{T} = \frac{1}{\pi} $. This tells us that the particle completes $ \frac{1}{\pi} $ cycles per second. Frequency lets us understand the temporal behavior of the oscillation.

Velocity Amplitude

The maximum speed at which the particle moves in SHM is represented by the **velocity amplitude**. It is given by the product of amplitude ($ A $) and angular frequency ($ \omega $). For our example: $ A = 4 $ and $ \omega = 2 $. Thus, $ \text{Velocity Amplitude} = A \cdot \omega = 4 \cdot 2 = 8 \, \text{units/second} $. This value helps us quantify the kinetic energy involved in the motion.