Pendulum connected to pulley system

[aaron.conway]

Imagine a standard pendulum. I understand how to work the physics on one but what if you attached that pendulum to a pulley at the top and could apply tension through the line? Let's say you released it at 45 degrees then some degrees later you applied some amount of tension through the pulley forcing the pendulum to both deviate from a circular path and increase in velocity. How could I come up with a function that showed where the pendulum would be with some tension applied and let go at some angle? The radius and weight of the pendulum would be fixed. Thanks for your time. I was thinking you could launch people even farther if you used this for a rope swing. They would swing then at some angle a tension would be applied and they would follow a more parabolic path and get launched.

Also, this is not a homework question please don't move it! I've been out of physics for a long time and was just looking for help in designing a new rope swing. I have my college physics book and am reading the crap out of it so please no negative comments.

 

[Haborix]

Interesting problem. You'll remember that the solution that we commonly have for the pendulum assumes that is swings through a small angle. However, there is a way to solve the pendulum problem for any angle. The solutions are given in terms of Jacobi elliptic functions. Your problem adds another layer which a force applied in the radial direction via the block and pulley. Now, I can't know for sure, but I suspect there isn't going to be an analytic solution for such a thing.

So, it would probably be easiest to use a numerical differential equation solver (pretty easy to implement with Mathematica) to get an idea of the solution before you go hunting for an analytic solution. I'll try to mess around with this before I go to bed tonight and see if anything sticks out.

Again, cool problem.

 

[aaron.conway]

Thanks for your input! I'll refresh on my dif-eq as well! The main thing I just want to be able to figure out is at what angle should I put the tension on the line to minimize the accelerative forces on the person yet maximize their horizontal distance. I think experimental testing would be easier than modeling an equation. Although it would be extremely useful if I wanted to replicate this idea else where. Thanks again and I'll make sure to post any work I do on it!

 

[aaron.conway]

Hey so incase anyone was wondering. The solution to my problem is called a swinging atwood machine. Mine would just be different in the fact that I would be using a force applied to the counter weight instead of gravity.