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

  • Thread starter carllacan
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In summary, the problem involves a pendulum with a length of 4a, placed between two obstacles described by parametrized coordinates x = a(θ+sinθ) and z = a(-3-cosθ). The obstacles affect the pendulum's natural movement. The goal is to find the equation of movement, and one approach is to treat the pendulum as a regular one with a changing length from L to L minus the portion of the obstacles the rope is in contact with. The obstacle is a cycloid and the parameter θ represents the angle over which the circle to describe the obstacle has rotated. Reading up on the cycloid can provide further insight into the problem.
  • #1
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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  • #2
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?
 
  • #3
I can note that the obstacle is a cycloid.

This might help.

http://www.17centurymaths.com/contents/huygens/horologiumpart1.pdf
 
  • #4
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.
 
  • #5
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...)
 

1. How does the presence of obstacles affect the motion of a pendulum?

The presence of obstacles can cause the pendulum to deviate from its expected path, leading to changes in its period and amplitude of oscillation. This is due to the fact that the obstacles introduce additional forces on the pendulum, altering its overall energy and momentum.

2. Can the length of the pendulum affect its interaction with obstacles?

Yes, the length of the pendulum can impact its interaction with obstacles. A longer pendulum will have a longer period of oscillation and therefore will take longer to complete a full swing, potentially allowing it to avoid or overcome obstacles. On the other hand, a shorter pendulum may be more easily affected by obstacles due to its shorter period and faster swings.

3. How do different types of obstacles affect the motion of a pendulum?

The type of obstacle can have a significant impact on the motion of a pendulum. For example, a solid obstacle may completely stop the pendulum's motion, while a series of smaller obstacles may cause it to gradually lose energy and come to a stop. In some cases, certain types of obstacles may even cause the pendulum to change direction or spin in a different plane of motion.

4. Is there a way to predict how a pendulum will interact with obstacles?

While there are equations and models that can help predict the motion of a pendulum in a vacuum, predicting its interaction with obstacles can be more complex and may require experimental data. Factors such as the mass and velocity of the pendulum, as well as the size, shape, and placement of the obstacles, can all play a role in determining the pendulum's behavior.

5. How can pendulums with obstacles be applied in real-world scenarios?

Pendulums with obstacles have a wide range of practical applications, such as in seismology to measure earthquakes, in clocks to keep time, and in amusement park rides for entertainment. They can also be used in experiments to study the effects of different forces and obstacles on motion, helping us better understand the laws of physics.

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