# Two Identical Pendulums Connected by a Light Spring

In summary, two identical pendulums connected by a light spring with displacements given by xa = Acos((w2-w1)t/2)cos((w2+w1)t/2) and xb = Asin((w2-w1)t/2)sin((w2+w1)t/2) have energies of Ea = 1/2 m*A^2*((w2+w1)/2)^2*cos^2((w2-w1)t/2) and Eb = 1/2 m*A^2*((w2+w1)/2)^2*sin^2((w2-w1)t/2). The book's solution assumes that the frequencies are similar, allowing the displacement equations to be viewed

## Homework Statement

Two identical pendulums of the same mass m are connected by a light spring. The displacements of the two masses are given, respectively, by xa = Acos( (w2-w1)t/2 )cos( (w2 + w1)t/2 ), xb = Asin( (w2-w1)t/2 )sin( (w2 + w1)t/2 ).

Assume that the sprint is sufficiently weak that its potential energy can be neglected and that the energy of each pendulum can be considered to be constant over a cycle of its oscillation.

Show that the energies of the two masses are are:

Ea = 1/2 m * A^2 ( (w2 + w1)/2 )^2 cos^2( (w2 - w1)t/2 )
Eb = 1/2 m * A^2 ( (w2 + w1)/2 )^2 sin^2( (w2 - w1)t/2 )

## Homework Equations

Energy of simple harmonic oscillator = (1/2)mw^2 (amplitude)^2

## The Attempt at a Solution

In the book's solution, it says that amplitude = A cos[ (w2 - w1)t/ 2] or A sin[ (w2 - w1)t/2 ]. Where does it get this from? What happened to the other cosine/sine term? Why is it using this particular term?

Last edited:
The displacements of the two masses are given, respectively, by xa = Acos( (w2-w1)/2 )cos( (w2 + w1)/2 ), xb = Asin( (w2-w1)/2 )sin( (w2 + w1)/2 ).
Those are constants. Should there be some occurrences of t in there?

Whoops, thanks for catching that. I edited my post.

If the two frequencies are similar, you can view the displacement equations as a rapid oscillation (the average of the frequencies) modulated in amplitude by a beat frequency (half the difference). In that view, we can treat the beat frequency factor as a time-dependent amplitude, A(t). Applying the standard formula should then yield the result.
That said, there is nothing in the statement of the question to justify the assumption that the frequencies are so close.

## 1. What is the purpose of connecting two identical pendulums with a light spring?

The purpose of connecting two identical pendulums with a light spring is to create a coupled pendulum system. This allows for the transfer of energy between the two pendulums, resulting in more complex and interesting motion patterns.

## 2. How does the length of the spring affect the motion of the coupled pendulum system?

The length of the spring affects the frequency of the coupled pendulum system. A shorter spring will result in a higher frequency of oscillation, while a longer spring will result in a lower frequency. This can also affect the amplitude and energy transfer between the two pendulums.

## 3. Is the motion of the two pendulums always synchronized when connected by a light spring?

No, the motion of the two pendulums may not always be synchronized. The degree of synchronization depends on the initial conditions and the length and stiffness of the spring. In some cases, the motion may become chaotic and the pendulums may not be synchronized at all.

## 4. Can the coupled pendulum system exhibit different types of motion?

Yes, the coupled pendulum system can exhibit different types of motion, such as in-phase motion where the two pendulums move together, anti-phase motion where the two pendulums move in opposite directions, and chaos where the motion is unpredictable. This depends on the initial conditions and the properties of the spring.

## 5. How does the addition of a light spring affect the period of the pendulums?

The addition of a light spring can affect the period of the pendulums by either increasing or decreasing it. This depends on the length and stiffness of the spring. In general, the period of the pendulums will be shorter when connected by a spring compared to when they are not connected.

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