Marbles on a meter stick

I talked with Peter Bohacek at the WebAssign Users Group Meeting (WAUG), and he showed me some of his excellent physics videos. One of them was this one below:

There are a number of questions that may come to mind. Two were of particular interest to me/us. The first is in looking at the final state, there are a number of balls that are in a straight line, and then a number that are at different heights. What determines the number of balls in the two parts? Secondly, why is the linear part not horizontal?

I wrote a VPython simulation for the process. The screencast below shows the output.

When I remove the code for the interaction between the balls and the meter stick, there were 7 marbles in the linear part. That makes sense, because the physical pendulum made of a meter stick by itself should fall with an angular acceleration that would be the same as a simple pendulum with length 2/3 meter. That means all of the balls at larger distances would accelerate more slowly than the meter stick. There are 7 such balls (1, 0.95, 0.9, 0.85, 0.8, 0.75, 0.7 meters). So those balls should instantaneously separate from the meter stick.

When the ball/stick interaction was added back in, the general shape of the falling balls was reproduced, as shown in the images below.

There are a few differences, though. While I used the correct masses and sizes of the objects, the simulation produced a linear section of balls that is one or two balls (depending on exactly which ones you count as in the linear section) bigger than the video. The simulation also gave a horizontal linear set of balls, while the experiment gave about a 3 degree incline.

One way to make the size of the linear parts match is to reduce the mass of the balls, thereby reducing the force they put on the meter stick and reducing its acceleration. I found that the mass would need to be reduced by over a factor of 2 from the reported values, which is not reasonable. Another is to increase the mass of the meter stick, but again a factor of 2 change in mass would be necessary. Adding some friction at the rotation point would also slow the rotation rate of the simulated meter stick, but (untested) that seems like it would be a too large amount to make the necessary change by itself.

I did video analysis on the experiment and recorded the motion of each ball and the stick itself. Something about the experiment causes a lower acceleration of the stick than the simulation has, which is the ultimate cause of the difference in linear section size. When comparing the falling balls, the simulation of ball at 100 cm matches the experiment quite well (small discrepancy in the free fall acceleration). The simulated ball at the 25 cm mark has the same shape as the experimental one, but it begins to fall sooner - again due to the larger acceleration of the stick itself.

Position of the stick as a function of time

Position of the balls at 100 and 25 cm as a function of time. 

The simulation was also unable to mimic the non-horizontal nature of the line of balls. It isn't clear physically where that characteristic would come from. At one point the simulation did have it, but I believe that was more an effect of the discrete motion of the simulation that caused the balls to touch, and then not touch the stick repeatedly. My current ball/stick interaction model only does horizontal lines of balls unless you start the stick at an angle, but then it just keeps the angle you started at - which is not the case in the video.

For the curious, here is a link to the source code for the simulation:  BallStick.py  I'd love to know why I can't match the experiment a little more closely.