Can you solve the pendulum with obstacles? Find the equation of movement!

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Homework Help Overview

The problem involves a pendulum of length 4a that is influenced by two rounded obstacles described by specific parametric equations. Participants are tasked with finding the equation of movement for the pendulum in this altered environment.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • The original poster expresses uncertainty about how to begin solving the problem. Another participant suggests treating the pendulum as a regular one but considers the changing length due to contact with the obstacles. There is a mention of the obstacles being cycloidal in nature, which may influence the pendulum's movement. Questions arise regarding the role of the parameter θ and its implications for the pendulum's motion.

Discussion Status

Participants are exploring various interpretations of the problem, with some suggesting potential approaches while others seek clarification on specific aspects. There is no explicit consensus yet, but ideas are being shared that may guide further exploration.

Contextual Notes

Participants are discussing the implications of the obstacles on the pendulum's movement and the parametrization involved. There is an acknowledgment of the complexity introduced by the obstacles, and the need for further understanding of the cycloidal nature of the obstacles is noted.

carllacan
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Homework Statement


We have a pendulum, with length 4a. It is placed between two rounded obstacles (see image) described by x = a(θ+sinθ) and z = a(-3-cosθ), which disturb its natural movement. Find the equation of movement.
https://www.dropbox.com/s/q1y4gzmnv0mm9c0/2014-01-27 20.19.03.jpg

Homework Equations


x = a(θ+sinθ) and z = a(-3-cosθ) are the parametrized coordinates of the obstacles


The Attempt at a Solution


I have no idea where to start.
 
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Okay, I've just had an idea for this, but I would appreciate if you gave me any opinion before I try to apply it.

I think I could treat the pendulum as a regular one, with the twist that its lentgh changues over time, from L to L minus the portion of the obstacles the rope is incontact with. Does that make any sense?
 
I can note that the obstacle is a cycloid.

This might help.

http://www.17centurymaths.com/contents/huygens/horologiumpart1.pdf
 
Not much, but thanky you.

I found an expression for the length of the rope not touching the obstacles, but I don't know what else to do.

Do you think the parameter θ is the polar coordinate of the mass? This would make things infinitely easier.
 
No, the introduction of theta is a parametrisation as you say. Basically the angle over which the circle to describe the obstacle has rotated. Work towards the angle between the tangent to the obstacle and the vertical.

And read up on the cycloid (what you found is OK, but it's a little verbose. In the years since 1673 shorter exposes were put on the net...)
 

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