31-May-2017: Conservation of Energy/Conservation of angular mom

Lab 19: Conservation of Energy/Conservation of angular mom

31-May-2017
Purpose: Demonstrate the conservation of angular momentum, by first using conservation of energy with a rod swiveling around a pivot, picking up a clay object at its midpoint.

Summary:

1) For this lab, we set up a clamp at the edge of the table, with a metal rod pointing out that would let us insert a meter stick through one of its pivots.

2) We put double sided tape at the end of the meter stick, so that when it starts to swing up again after released from rest, it would grab a clay target placed at the bottom of its arc. 

3) Directly facing the apparatus, we stationed a mobile phone to capture the process at 60 fps.

4) Once the video was captured, we took the footage into LoggerPro, plotted the points, and obtained the experimental value for the final height of the clay blob by using its "scale" operation to determine the height of a point placed directly over the clay.

This was our set up as seen in LoggerPro:

5) After that, it became a matter of deriving my various expressions for energy and angular momentum in order to calculate the theoretical value of the clay's height!

Analysis

1) LoggerPro gave us a value for the blob's actual height, which we will use as the experimental value with which to compare our calculated, our theoretical, result. In order to do the energy calculations that would give me this value for height, we first needed to know three things:

1. When released from rest, what angular velocity does the meter stick have?
2. After it makes contact with the blob, what is its inertia?
3. Once it begins to rise again, what is its new angular velocity?

Number 1 allows us to solve for number 3 by letting us set up an expression for angular momentum. For number 2, because the meter stick is by itself for now, we use the expression for the parallel axis theorem to solve for the inertia of this stick when its pivot is 10 cm from the end. Plugging this into our expression for the stick's angular momentum, we arrive at a value for the stick's initial angular velocity for Number 1.

Then, we needed to know what angular velocity our system had following the collision, which we set up exactly the same as Number 1, but with the expression for inertia we derived using the parallel axis theorem, PLUS the inertia of the blob that is a distance away from the pivot. We got a value that was significantly less than the angular velocity before the collision, which is expected.

Finally, we could set up our expression for conservation of energy to solve for the height of the clay. The system starts with a kinetic angular velocity, and ends with a potential energy at some height. However, we took note that it was important not to forget about the potential energy of the stick; so we had two variables for h on the right: the height of the stick's center mass, and the height of the blob. Because the stick is rotating, we used the energy expression for the stick-blob system. Then it became a matter of isolating my h_clay variable, and solving for it.

Having arrived at my theoretical value, percent error reveals that my calculations were reasonable, but had some important uncertainties, at 10.9%.

Measured Data

M_stick =  0.141 kg

m_blob = .021 kg

H_exp = .4003 m

Calculated Data

W_0 = 5.67 rad/s

W_f =  3.79 rad/s

H_clay_theoretical = 0.449 m
|% Error| = 10.9%

Conclusions

Based on how high my percent error was, the sources of uncertainty here clearly had a significant impact on my final results. The first of which was no doubt the fact that in real life, no collisions are perfect, and the stick+clay collision results in a loss of energy that leads it to the stick+blob not making it up as high as its theoretical value shows it should be.

Our set up was not ideal either, as energy is realistically lost due to friction where the meter stick's hole meets the pivot, again, resulting again in a loss of energy that leads to the stick+blob not making it up as high as it should. 

Another important assumption we glossed over was how we treated the blob as a point mass at the end of the stick, when in reality, it was an amorphous object likely with a moment of inertia in accordance with its own shape, an error that likely makes our prediction for the theoretical height of the blob too big.

In our calculations, the propagated uncertainties with this lab were minimal, such as the mass of our objects. What could possibly be more significant however, is LoggerPro. As previously explained, we used LoggerPro's "scale" tool to let the program know that the length of the stick was 1 meter. However, the camera shot the footage from the ground, looking up some angle toward the stick. Therefore LoggerPro has an inaccurate understanding of the actual scale of our system, which could explain that our experimental value of height was simply the fault of scaling.

However, based on how my calculations ultimately did compute, and return reasonable numbers, it is safe to assume that the calculations above reflect what is going on in the apparatus in terms of conservation of energy and conservation of momentum.