## Can a pendulum save you?

I’m so thankful to my friend Chija for pointing out this video for me:

Here’s her tweet

When I saw it I started to wonder if angular momentum was enough to explain it. So I set about trying to model it. Here’s my first try: Green ball is 20x the mass of the red. No contact or air friction.

It does a pretty good job showing how the fast rotation of the red ball produces enough tension in the line to slow and then later raise the green ball. Here’s a plot of the tension in the line as a function of time: Tension in the line as a function of time. The green line is the strength of gravity. The reason everything is negative is a consequence of how I modeled the constraint (a Lagrange multiplier)

So how did I model it? I decided to use a Lagrange multiplier approach where the length of the rope needs to be held constant. Here’s a screenshot of the code: “ms” is a list of the masses. “cons” is the constraint.

You define the constraint, the kinetic and potential energies, and then just do a lagrangian differential equation for x and y of both particles: $\frac{\partial L}{\partial x}-\frac{d}{dt}\frac{\partial L}{\partial x'}+\lambda(t)\frac{\partial \text{cons}}{\partial x}=0$

(note that in the screen shot above there’s actually some air resistance added as an extra term on the left hand side of the “el” command).

Very cool. But what about the notion that the rope wraps around the bar, effectively shortening the string? I thought about it for a while and realized I could approach the problem a little differently if I used radial coordinates. First here’s a code example of a particle tied to a string whose other side is tied to the post: “rad” is the radius of the bar. Note how the initial “velocities” of the variables need to be related through the constraint.

I’ve changed the constraint so that some of the rope is wrapped around the bar according to the angle of the particle. Here’s what that yields: Ok, so then I wanted to feature wrapping in the code with both masses. Here’s that code: Note the negative sign before “l[t]” and the “ $\theta[t]$” in the constraint.

And here’s the result, purposely starting the more massive object a little off from vertical: Fun times! Your thoughts? Here are some starters for you:

• Why do you insist on using Mathematica for this? It would be much easier in python, here’s how . . .
• Some of the animations don’t look quite right to me. Are you sure that . . .?
• This is cool, do you plan to do this for your students soon?
• What about contact friction between the rope and the bar? I would think that would be a major part.
• In the video he just comes to a rest instead of bouncing up. Clearly you’ve done this all wrong. Professor of physics at Hamline University in St. Paul, MN
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### 3 Responses to Can a pendulum save you?

1. Steve says:

Love this video. Such a cool effect just begs to be analyzed using physics principles.

I’ve noticed that you have taken a Lagrangian approach to this problem, but I teach intro physics. I envision a PBL session centered around the question “Using your physics knowledge, explain why the daredevil in this video doesn’t die.”

I could also see this: “Your local SPS chapter wants to recreate the experiment done in this video, and your faculty advisor has encouaged you to study this problem so that you remain safe.” Then ask them to generate code to answer at least one of the following questions.
1. What is the minimum counterweight mass that is required?
2. What is the minimum height of drop that will allow you to remain safe? (We want the rope mechanism to stop you, not the ground!)
3. What is the required tensile strength of the rope to ensure safety.

• Andy "SuperFly" Rundquist says:

I did a Lagrangian approach because it got me to the equations of motion the fastest. But really, they’re all just a bunch of F=ma equations that any student should be able to get from a nice free-body diagram approach. The problem is that none of the tension forces are constant and so that’s why I needed a computational solution.