Lab 19: Angular Momentum

If a ruler held from a fixed pivot swings down and hits a mass of putty, how high can it continue to swing? Or alternatively, how high can this shoe swing after picking up that ball of gum?

Or how wide can my mood swing after being denied this sandwich because the school store had a line and there was only one worker there?

These are things that may or may not involve angular momentum. For this lab, we will try to analytically predict the swing of the ruler, and eat that sandwich while no one's looking. Then, we will compare our predictions with results from a video capture. In order to figure this out expediently, I swiftly weighed our ruler and putty. We need to find the length of the meter stick... before the pivot.

So we used the ruler to measure itself recursively, and pondered whether it would be possible for it to measure it measuring itself, and so on... However, that idea had to be put to rest, for the sooner we set up the experiment is when I get to pass off the tough stuff to my math wiz lab partner and eat my sandwich, killing two physics problems with one putty. Fortunately, my appetite was sated. Unfortunately, my lab partner did not do the correct math the first time. In any case, I present our measurements:

Measurements
Axis shift from center of mass to pivot	0.493 m
Mass of ruler	0.088 kg
Mass of putty	0.015 kg

There are three steps to this problem. First, as the ruler is held up and starts its descent, gravitational potential energy converts into rotational kinetic energy. At the bottom of the swing is when the ruler moves the fastest, since all of the energy has been converted. Second, when the ruler hits the putty, angular momentum is conserved; however, since it's an inelastic collision, the angular speed changes. The lower speed clearly lowers the energy that could be converted back into gravitational potential. So our three steps are:

1. GPE → rot KE
2. L₀ = L_f
3. rot KE → GPE

The moment of inertia of the ruler is as a bar rotating about its center of mass, but shifted to one side (using the parallel axis theorem). Because the ruler is not swing exactly at an edge, it would be more reasonable to use the inertia formula for swinging about the center than swinging about an edge. We begin converting energy to figure out the speed of the ruler just as it reaches the bottom:

Then, we use angular momentum to figure out the speed of the ruler just after it makes contact with the putty:

Now we find the height from the kinetic energy of the ruler converted back into gravitational potential:

The numbers are plugged in. Voila!

To be honest, that seems a bit high, but we are assuming that there is no rotational friction in the pivot or air resistance, so it probably makes sense that the numbers do not match personal experience. There is usually friction, whether between objects or air, whenever an object swings about another, which is why a playground swing does not act as an ideal pendulum. How convenient would that be for the parents! Push once, and, "I'll be back in half an hour, son!" Anyways, now to set up and do this experimentally:

The clay is wrapped with some tape to aid it in sticking on the ruler.

There are only power strips on one side of the table, and we didn't want to move the entire set up, so we had to do some creative ad hoc wiring...

We use a video capture to record the swing of the ruler. Through a string of bad luck, we could never set up the camera correctly. For one, the preview didn't work, so we couldn't check the clarity beforehand. Second, higher definition seemed to cause problems. Maybe I should bring my own camera to physics classes next year!

As you can see, Logger Pro shows a paltry 0.1321 m for the final height, about a third of our analytic calculations. We should've brought some Viagra! If we count the blurry dots on the meter stick though, which presumably represent 5 or 10 cm marks, the blue dot waaay to the right is about 3-4 notches up, suggesting that the scale is probably calibrated badly. Also, since the capture was so blurry and overexposed, we had to make some guess-timates on where the end of the ruler was. These problems could contribute to the disparity, but I still think that friction isn't eliminated in the pivot. If we lubed it up, the stick would've probably gone higher. Just saying!