What are the simple modes and frequencies for weakly coupled oscillators?

Click For Summary

Homework Help Overview

The problem involves two simple pendulums of equal length connected by a spring, with a focus on the oscillatory energy exchange between them when one is displaced. The inquiry centers around determining the period of this energy transfer and understanding the modes and frequencies of the system.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the relationship between the amplitudes of the pendulums during oscillation, referencing energy conservation and the concept of "beat" frequency. There are attempts to derive equations related to the timing of maximum amplitudes and questions about relating these findings to the period of energy transfer.

Discussion Status

The discussion is active, with participants exploring different aspects of the problem. Some have provided insights into the relationship between the pendulums' motions, while others are questioning how to connect their mathematical findings to the concept of energy transfer periods. There is no explicit consensus yet on the modes or frequencies of the system.

Contextual Notes

Participants are navigating the complexities of the problem, including potential assumptions about the system's behavior and the nature of the coupling between the pendulums. There is mention of the context possibly relating to differential equations or Lagrangian mechanics, but no definitive conclusions are drawn.

Zanatto
Messages
2
Reaction score
0

Homework Statement


Two simple pendulums os equal length L=1m are connected with spring with a spring constant K=0,05 Mg/L. The pendulums are started by realeasing one of them from a displaced position. The subsequent motion is characterized by an oscillatory energy exchange between the pendulums. What is te period of this transfer?

Homework Equations

The Attempt at a Solution


In this situation, as a pendulum is displaced and the another is static, I suppose that when the amplitude of pendulum 1 is a maximum, the amplitude of pendulum 2 is minimum, because the result is displaying "beat" frequency. That's correct?
 
Physics news on Phys.org
Yes. Why ask ?
 
  • Like
Likes   Reactions: Zanatto
Yes. When the amplitude of one of the pendula is zero, the amplitude of the other is maximal. This follows directly from energy conservation.
 
  • Like
Likes   Reactions: Zanatto
Well, just to get rid of the doubt.

Them, starting from the equation of position, and I found that the maximum values of the amplitude for the pendulum 1 occurs when:

επ(t/T) = nπ ---> t/T = n/ε

The time between maxima is T/ε, inversely proportional to the coupling spring constant.

And for the pendulum 2 when:

επ(t/T) = (2n +1)* π/2 ---> t/T = (2n+1)/2ε

Them, I stuck in here. Can I relate that with period of transfer in the oscillatory energy exchange in some way?
 
Good. So what's this exercise ? In a chapter on differential equations, on Larangian mechanics, a lab instruction preparation perhaps ?

The motion described is asymmetric and one can expect the pendulum that's initially immobile to start swingning too.
There are simple modes possible where there is no transfer; can you guess which ? Wht are their frequencies ?
 

Similar threads

Replies
2
Views
2K
  • · Replies 1 ·
Replies
1
Views
1K
Replies
4
Views
2K
Replies
5
Views
4K
  • · Replies 16 ·
Replies
16
Views
2K
  • · Replies 7 ·
Replies
7
Views
2K
  • · Replies 3 ·
Replies
3
Views
4K
Replies
3
Views
1K
  • · Replies 10 ·
Replies
10
Views
3K
  • · Replies 3 ·
Replies
3
Views
2K