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?

Step by step solution

01

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: \(\omega_{a}=\left(\omega_{2}+\omega_{1}\right) / 2\), \(\quad\omega_{b}=\left(\omega_{2}-\omega_{1}\right) / 2\)

02

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 \(A_2(t)\): \(A_{1}(t) \cong \theta_{0} \cos \omega_{b} t\) \(A_{2}(t) \cong \theta_{0} \sin \omega_{b} t\)

03

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: \(\theta_{1}(t) \cong A_{1}(t) \cos\omega_{a}t = \theta_{0} \cos \omega_{b} t \cos \omega_{a} t\) \(\theta_{2}(t) \cong A_{2}(t) \sin\omega_{a}t = \theta_{0} \sin \omega_{b} t \sin \omega_{a} t\)

04

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.

05

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.

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