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

[carllacan]

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.

 

[carllacan]

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?

 

[j824h]

I can note that the obstacle is a cycloid.

This might help.

http://www.17centurymaths.com/contents/huygens/horologiumpart1.pdf

 

[carllacan]

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.

 

[BvU]

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...)