Question

Consider the special case where the pendulum system of Problem \(7.3\) has the property that \(k / m \ll g / L\). If the pendulums are released from rest when in the configuration \(\theta_{1}(0)=\theta_{0}\) and \(\theta_{2}(0)=0\), show that the response is of the form $$ \begin{aligned} &\theta_{1}(t) \cong A_{1}(t) \cos \omega_{a} t=\theta_{0} \cos \omega_{b} t \cos \omega_{a} t \\ &\theta_{2}(t) \cong A_{2}(t) \sin \omega_{a} t=\theta_{0} \sin \omega_{b} t \sin \omega_{a} t \end{aligned} $$ $$ \omega_{a}=\left(\omega_{2}+\omega_{1}\right) / 2, \omega_{b}=\left(\omega_{2}-\omega_{1}\right) / 2 $$ where $$ \omega_{a}=\left(\omega_{2}+\omega_{1}\right) / 2, \quad \omega_{b}=\left(\omega_{2}-\omega_{1}\right) / 2 $$ Plot the response. What type of behavior does the pendulum system exhibit?

Short answer

Answer: The behavior of the two coupled pendulums is a combination of two sinusoidal motions, one at the average frequency (ω_a) and the other at the difference frequency (ω_b). This motion is typical for a coupled oscillator with weak coupling, indicating that the motion of the two pendulums is mainly independent but mixed with a small component of the other oscillation.

Step-by-step solution

Identify the terms \(\omega_1\) and \(\omega_2\)

We can assume that \(\omega_1\) and \(\omega_2\) are the natural frequencies of the two pendulums. We can write \(\omega_1 = \sqrt\frac{g}{L_1}\) and \(\omega_2 = \sqrt\frac{g}{L_2}\), where \(L_1\) and \(L_2\) are the lengths of the two pendulums. Next, we need to find the values for \(\omega_a\) and \(\omega_b\) using the given formulas: \(\omega_{a}=\left(\omega_{2}+\omega_{1}\right) / 2\), \(\quad\omega_{b}=\left(\omega_{2}-\omega_{1}\right) / 2\)

Find \(A_1(t)\) and \(A_2(t)\) for the given conditions

Since the exercise states that \(k / m \ll g / L\), we can assume that the damping effect due to \(k\) (the spring constant) will be negligible. Under this condition, we can use simple harmonic motion equations to find \(A_1(t)\) and \(A_2(t)\): \(A_{1}(t) \cong \theta_{0} \cos \omega_{b} t\) \(A_{2}(t) \cong \theta_{0} \sin \omega_{b} t\)

Write the equations for pendulum responses

Using the values we've found for \(A_1(t)\) and \(A_2(t)\), we can now write the equations for the pendulum responses: \(\theta_{1}(t) \cong A_{1}(t) \cos\omega_{a}t = \theta_{0} \cos \omega_{b} t \cos \omega_{a} t\) \(\theta_{2}(t) \cong A_{2}(t) \sin\omega_{a}t = \theta_{0} \sin \omega_{b} t \sin \omega_{a} t\)

Plot the response

You can use a graphing calculator or software to plot the responses with the given equations for \(\theta_1(t)\) and \(\theta_2(t)\). Make sure to plug in the appropriate values for \(\theta_0\), \(L_1\), and \(L_2\) given by the problem.

Describe the behavior

Observing the plotted graphs, we can see that the pendulum system exhibits a combination of two sinusoidal motions, one at the average frequency \(\omega_a\) and the other at the difference frequency \(\omega_b\). This behavior is typical for a coupled oscillator with weak coupling, showing that the motion of the two pendulums is mainly independent but mixed with a small component of the other oscillation.

Key concepts

Harmonic Motion

Harmonic motion is a type of periodic motion where an object oscillates back and forth in a regular pattern. A simple example is the swinging of a pendulum. In the context of the pendulum system given in the exercise, it undergoes harmonic motion when it swings from one side to another. The frequency at which it oscillates is determined by factors such as the length of the pendulum, and the acceleration due to gravity.
To identify harmonic motion in mathematical form, it usually involves trigonometric functions such as sine or cosine. For instance, the motion of the pendulum can be expressed as \( \theta(t) = A \cos \omega t \), where \( \theta(t) \) is the angular displacement at time \( t \), \( A \) is the amplitude, and \( \omega \) is the angular frequency. This expression shows how the motion is repetitive and predictable.
The simple harmonic motion equations are used to simplify the understanding of the movements of the pendulum, especially under conditions where damping forces (like air resistance) can be neglected or are minimal.

Natural Frequencies

Natural frequencies are the specific frequencies at which a system tends to oscillate when not being driven or damped by an external force. In a pendulum system, these frequencies depend on the physical parameters of the system like length and gravity.
For a pendulum, the natural frequency can be calculated using the formula \( \omega = \sqrt{\frac{g}{L}} \), where \( g \) is the acceleration due to gravity and \( L \) is the length of the pendulum. This formula helps to understand how quickly the pendulum swings back and forth.

• \( \omega_1 = \sqrt{\frac{g}{L_1}} \) and \( \omega_2 = \sqrt{\frac{g}{L_2}} \) are the natural frequencies of the two pendulums given different lengths.
• Calculating these allows us to derive new terms such as \( \omega_a \) and \( \omega_b \), where \( \omega_a = \frac{\omega_2 + \omega_1}{2} \) and \( \omega_b = \frac{\omega_2 - \omega_1}{2} \).

These terms play an important role in understanding oscillations in more complex systems such as when pendulums are connected or coupled.

Coupled Oscillators

Coupled oscillators are systems where two or more oscillators are linked in such a way that they influence each other. This is precisely what happens in the given pendulum system when it is assumed that \( k/m \ll g/L \), indicating weak coupling.
In coupled oscillators, each pendulum has its natural frequency, but they also affect one another's motion through coupling. In mathematical terms, the response of each pendulum can be described with equations incorporating elements of both oscillators.

• The first pendulum's motion, \( \theta_1(t) \), which is linked to the oscillation of the second and vice versa, is shown as a combination of two cosine functions.
• Similarly, the second pendulum, \( \theta_2(t) \), shows a combination of sine functions.

This behavior exhibits an interesting dynamic known as "beating," where the energy periodically transfers between the two pendulums at the difference frequency \( \omega_b \). This results in an observable pattern of amplitude modulation in their swings.

Sinusoidal Motion

Sinusoidal motion describes any motion that follows a sine or cosine waveform. This type of motion is smooth and periodic, meaning it repeats at regular intervals. When analyzing pendulums, sinusoidal motion comes into play as it describes how the pendulums swing.
In the exercise, the pendulum's motion is expressed as a product of sinusoidal functions:

• The first pendulum's angle \( \theta_1(t) \) includes terms \( \cos \omega_b t \) and \( \cos \omega_a t \), showing a primary oscillation modified by another oscillation.
• The second pendulum's response \( \theta_2(t) \) similarly involves sine functions like \( \sin \omega_b t \) and \( \sin \omega_a t \).

These combinations are integral for understanding the complex patterns of coupled oscillations. The oscillations described by \( \omega_a \) and \( \omega_b \) create shifts that are crucial for accurately capturing the pendulums' behavior. Sinusoidal motion also illustrates how the amplitudes of oscillations may vary over time due to interactions between the coupled pendulums, leading to scenarios such as rhythmical "beats" in the system.