What Determines the Maximum Length of a Spring in an Elastic Pendulum?

[MusNothus]

Homework Statement

Originally I was considering this question:
"You have a pendulum with a ball of mass m at the end, and the pendulum is held parallel to the ground at equilibrium length so there is no initial spring force. This pendulum is also a spring, however, with spring constant k and equilibrium length L. You let go of the weight. What is L', the new length of the spring at the bottom of the swing (pi/2), and what is the velocity of the ball?"

Initially I assumed that at the bottom of the swing the velocity vector would be perpendicular to the acceleration vector, which would be parallel with the spring. Therefore I tried using uniform circular motion equations to solve the problem and assumed that the length at the bottom of the swing was the maximum spring length. I wasn't too certain that this was a safe assumption, however, so I tested it experimentally.

I found that the spring actually didn't reach maximum length until after pi/2 radians. I talked to my professor about this and he had another professor use a simulation program to use 2nd order DEs in polar coordinates to solve the parametric path of the mass. This supported my experimental results, and the image is below if it shows up (radius vs. time, the angle vs. time and the parametric path):

[URL=http://img147.imageshack.us/my.php?image=bungeeck9.jpg][PLAIN]http://img147.imageshack.us/img147/4649/bungeeck9.th.jpg[/URL][/PLAIN]

The thing that interests me now is what determines the angle at which the spring reaches its maximum length and what is the conceptual or mathematical reason for it. I'm very curious about this question and I'd greatly appreciate any insight.

Homework Equations

about as far as I can get is:

velocity vector = radius dot * radius unit vector + radius * theta dot * theta unit vector
AND
acceleration vector = radius dot dot * radius unit vector + 2 * radius dot * theta dot * theta unit vector + radius * theta dot dot * theta unit vector - radius * theta dot squared * radius unit vector

but those could be wrong. After this you'd need to do 2nd order DEs.

The Attempt at a Solution

My intuitive hypothesis would be that a larger spring constant or lighter weight would make the angle smaller and a weaker spring constant or heavier weight would make the angle larger. It also seems that this angle could not be any smaller than pi/2. I've thought about conceptual reasons for these conclusions and haven't been able to come up with any support for them. I also am not familiar enough with the math to mathmatically derive a solution and I don't have access to the simulation they used so I can't just run a bunch of different masses and spring constants.

 

[Jhero]

Thank you for sharing your findings and hypotheses with us. I am always interested in exploring and understanding phenomena such as the one you have described. After reading your post and examining the simulation image, I have a few thoughts and suggestions that may help in further understanding this system.

Firstly, I would like to address the assumption you made about the maximum length of the spring being reached at the bottom of the swing. While this may seem like a logical assumption, it is important to remember that the pendulum is experiencing both gravitational and spring forces, which can affect its motion. In this case, the spring force is not constant, as it changes with the displacement of the pendulum. This means that the maximum length of the spring may not necessarily coincide with the bottom of the swing.

Secondly, I would suggest looking into the concept of simple harmonic motion, which is the motion of a system that experiences a restoring force proportional to its displacement from equilibrium. This is applicable to your pendulum-spring system, as the spring force is proportional to the displacement of the pendulum. By understanding this concept, you may be able to mathematically derive a solution for the angle at which the spring reaches its maximum length.

Finally, I would recommend conducting further experiments with different masses and spring constants to see if your hypothesis holds true. This will not only provide more data to support your findings, but it may also help in identifying any other factors that may affect the behavior of the pendulum-spring system.

I hope this helps in your exploration of this interesting system. Keep up the good work and don't hesitate to reach out for further discussion or assistance. Best of luck in your studies!