Two pendulums connected with a massless rope

In summary, the problem discusses two pendulums of equal mass and length that oscillate in the same horizontal plane and are connected by a massless horizontal rope. The goal is to determine the dependence of the pendulums' amplitudes on time. The equations for harmonic oscillation, kinetic energy, and potential energy are provided, but it is unclear how to use them to find the amplitude dependence. The solution involves using the Lagrangian and small angle approximation to obtain the equation for simple harmonic motion with angular frequency ω2=g/l. As the pendulums are connected by a massless rope, the dynamics reduce to a 1 degree of freedom system, where ##\theta_1=\theta_2##. However
  • #1
LagrangeEuler
717
20

Homework Statement


Two pendulums of same mass and length that oscillate in same horizontal plane are connected with maseless horizontal rope. What is dependence of amplitude of pendulums as a function of time?

Homework Equations


For harmonic oscilation
[tex]x=x_0\sin(\omega t+\varphi_0) [/tex]
Kinetic energy
[tex]E_k=\frac{1}{2}mv^2[/tex]
Potential energy
[tex]E_p=\frac{1}{2}kx^2[/tex]

The Attempt at a Solution


In case from the problem I suppose that kinetic energy is simple
[tex]E_k=E_{k1}+E_{k2}=\frac{m}{2}(\dot{\varphi}_1^2+\dot{\varphi}_2^2)l^2[/tex]
and potential energy is
[tex]E_p=E_{p1}+E_{p2}=mgl(1-\cos\varphi_1)+mgl(1-\cos \varphi_2)[/tex]
However I am not sure how from this to get amplitude dependence of pendulums as a function of time.
 
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  • #2
LagrangeEuler said:

Homework Statement


Two pendulums of same mass and length that oscillate in same horizontal plane are connected with maseless horizontal rope. What is dependence of amplitude of pendulums as a function of time?

Homework Equations


For harmonic oscilation
[tex]x=x_0\sin(\omega t+\varphi_0) [/tex]
Kinetic energy
[tex]E_k=\frac{1}{2}mv^2[/tex]
Potential energy
[tex]E_p=\frac{1}{2}kx^2[/tex]

The Attempt at a Solution


In case from the problem I suppose that kinetic energy is simple
[tex]E_k=E_{k1}+E_{k2}=\frac{m}{2}(\dot{\varphi}_1^2+\dot{\varphi}_2^2)l^2[/tex]
and potential energy is
[tex]E_p=E_{p1}+E_{p2}=mgl(1-\cos\varphi_1)+mgl(1-\cos \varphi_2)[/tex]
However I am not sure how from this to get amplitude dependence of pendulums as a function of time.[/B]
The lagrangian is KE=ml21dot22dot2)/2
PE=mgl(θ1222)/2+Constt.for small angle approximation
Solving this for θ1 and θ2 will yield the same equation for a SHM with angular frequency ω2=g/l
And the angular displacement would be θ1~Asin(ωt) and θ2~ Bsin(ωt)
 
  • #3
Apashanka said:
The lagrangian is KE=ml21dot22dot2)/2
PE=mgl(θ1222)/2+Constt.for small angle approximation
Solving this for θ1 and θ2 will yield the same equation for a SHM with angular frequency ω2=g/l
And the angular displacement would be θ1~Asin(ωt) and θ2~ Bsin(ωt)

If they are connected by a massless rope, then surely the dynamics reduces to a 1 degree of freedom system. ##\theta_1=\theta_2##.
 
  • #4
It is not that simple I think. Rope has constant length. So what is happening if pendulums go to opposite directions?
 

Related to Two pendulums connected with a massless rope

1. What is the purpose of connecting two pendulums with a massless rope?

The purpose of connecting two pendulums with a massless rope is to create a system where the motion of one pendulum affects the motion of the other. This allows for the study of coupled oscillations and the properties of a system with multiple pendulums.

2. How does the length of the rope affect the motion of the pendulums?

The length of the rope affects the motion of the pendulums by determining the period of oscillation. A longer rope will result in a longer period of oscillation, while a shorter rope will result in a shorter period. This is because the length of the rope affects the distance that the pendulum can swing, which in turn affects the time it takes for the pendulum to complete one full swing.

3. What factors affect the synchronization of the pendulums?

The synchronization of the pendulums is affected by several factors, including the length of the rope, the mass of the pendulums, and the initial displacement of the pendulums. These factors determine the natural frequency of each pendulum and how they interact with each other.

4. Can the pendulums ever become perfectly synchronized?

In theory, the pendulums can become perfectly synchronized if all external factors are eliminated and the pendulums have identical natural frequencies. However, in reality, there will always be slight differences in the pendulums' motions due to external influences, making perfect synchronization impossible.

5. How does the motion of the pendulums change when the rope is replaced with a rigid rod?

When the rope is replaced with a rigid rod, the motion of the pendulums changes from a coupled oscillation to a simple harmonic motion. This is because the rigid rod prevents the pendulums from interacting with each other, and each pendulum will oscillate independently with its own natural frequency.

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