Coupled pendulums problem

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In summary, the conversation discusses the use of two coupled pendulums at positions x,1 and x,2. Newton's equations for the forces result in two equations, which lead to two solutions. These solutions can be rewritten in a more interesting form using the equations for sinA ± sinB and cosA ± cosB. The person asking for help is stuck after expanding the brackets and is looking for guidance on how to proceed.
  • #1
8614smith
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Homework Statement


two coupled pendulums are used at positions x,1 and x,2

Newton’s equation for the forces leads to the two equations:

m,1 * (second derivative of x,1 with respect to t) = -k,1x,1 + k(x,2 - x,1)

and m,2 * (second derivative of x,2 with respect to t) = -k,2x,2 - k(x,2 - x,1)

This leads to the two solutions:

x,1(t) = A,1*sin(ω,1*t + α,1) + A,2*sin(ω,2*t + α,2) (equation 3)

and

x,2(t) = A,1*sin(ω,1*t + α,1) - A,2*sin(ω,2*t + α,2) (equation 4)

where

A,1 = A,2 and α,1 = α,2

rewrite equations (3) and (4) in the very interesting form:

x,1(t) = 2A,1*cos(((ω,1 - ω,2)/2)*t)sin(((ω,1 + ω,2)/2) (equation 3a)

and

x,2(t) = 2A,1*sin(((ω,1 - ω,2)/2)*t)cos(((ω,1 + ω,2)/2) (equation 4a)



Basically i have to derive (3a) and (4a) from equations 3 and 4 using A,1 = A,2 and α,1 = α,2

Homework Equations



above

The Attempt at a Solution



all I've managed to do is expand out the brackets and that's where i get stuck, is there anyone that can help get me in the right direction as i have no idea where to go from here or how it changes from sin to a cos,

thanks
 
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  • #2
Welcome to PF!

Hi 8614smith! Welcome to PF! :smile:

You need to learn the four equations for sinA ± sinB and cosA ± cosB.

In this case, use sinA + sin B = 2.sin((A+B)/2).cos((A-B)/2) :wink:
 

1. What is a coupled pendulums problem?

A coupled pendulums problem is a mathematical and physical problem that involves two or more pendulums that are connected to each other through a common point or by a coupling mechanism. The behavior and motion of each pendulum is affected by the motion of the other pendulums in the system.

2. How is the motion of coupled pendulums described?

The motion of coupled pendulums is described by a set of coupled differential equations, known as the coupled pendulums equations. These equations take into account the mass, length, and initial conditions of each pendulum, as well as the coupling between them.

3. What are the applications of coupled pendulums?

Coupled pendulums have various applications in physics and engineering. They can be used to model and study the behavior of systems that involve coupled oscillators, such as molecular vibrations, electrical circuits, and even the synchronization of fireflies. Coupled pendulums are also commonly used in educational demonstrations to illustrate concepts of resonance and energy transfer.

4. What factors affect the behavior of coupled pendulums?

The behavior of coupled pendulums is affected by various factors, including the length and mass of each pendulum, the initial conditions of the system, and the strength of the coupling between the pendulums. Other external factors, such as air resistance and friction, can also affect the motion of the pendulums.

5. Can coupled pendulums exhibit chaotic motion?

Yes, coupled pendulums can exhibit chaotic motion under certain conditions. Chaotic motion refers to a type of unpredictable and irregular motion that is highly sensitive to initial conditions. This can occur in coupled pendulums when the coupling strength is high and/or the initial conditions are very similar.

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