07-June-2017: Physical Pendulum Lab

Physical Pendulum Lab

07-June-2017
Purpose: Predict the period of two pendulums by deriving an expression for the inertia of a semicircle and an isosceles triangle. Then, verify the results experimentally.

Summary:

1) Before class, we went and derived the various expressions of the moment of inertia of an isosceles triangle and a semicircle at various pivots parallel to the lamina by first finding their center mass, and then utilizing the parallel axis theorem. In order to get an expression for the period, it was a matter of finding their various angular frequencies.

2) Next we set up a stand with a clamp fixed perpendicular, from which we hung a triangle and semicircle at a pivot fixed about the peak of the triangle, and the base-midpoint of the semicircle, after measuring their dimensions. We hung them from the rod by taping two paper clips to the tops. In addition, taking their mass was not necessary, as it is was variable that cancels out in our calculations.

3) Because we wanted to record the period of its harmonic motion once we gave it a push, we taped a piece of paper running down the middle of our two objects, to trigger a photogate as the paper swung through its motion.

4) Once we got our data from Logger Pro, we compared our experimental results with our calculated theoretical results.

Analysis

To ultimately predict the period of our two pendulums, we would have to find the moment of inertia about our fixed pivot points for the triangle, with base B height H, and the semicircle with radius R.

To do this in the first place, we have to know how far these pivots are from the objects' center mass. For both of our objects, I placed them base-down, with their x-midpoint at the origin, so that our shapes were symmetrical about the x-axis. This meant that we only cared about their Y center of mass, which I did by deriving an expression for the triangle and the circle, and plugging them into our known expression for the Y_cm.

Then, I derived expressions for its inertia along and edge.

• For the semicircle, I found the inertia at the midpoint of its base - and using the parallel axis theorem, solved for the object's inertia about its center of mass.
• For the triangle, I found the inertia along the pivot on its peak, and again, using the parallel axis theorem, solved for the object's inertia about its center of mass.

In addition, I went ahead and derived expressions for the angular frequency, which we would later need to calculate for our period.

Then, I solved for the inertia of our objects at our last pivot points.

• For our semicircle, we found its inertia about the point directly opposite the midpoint of the base, at the circle's outer rim using the parallel axis theorem.
• For our triangle, we found its inertia about the midpoint of its base, again using the parallel axis theorem.

Again, I went ahead and derived expressions for the angular frequency, which we would later need to calculate for our period.

After running our experiment and gathering our data from LoggerPro for our triangle rotated about its peak and the semicircle about the midpoint of its base, we plugged in our calculations for the shapes' various dimensions into our expressions for angular frequency, and subsequently for period.

Doing so, we got values with excellent accuracy, and when compared to our experimental results from LoggerPro, for the triangle, 1.3%, and for the semicircle, 2%.

Measured/Data:

Base of triangle: .205 m

Height of triangle: .227 m

Radius of circle: .152 m

Period of the triangle

Period of the semicircle

Calculated Data:

T_triangle: 0.856 s

T_circle: 0.849 s

|%Error Triangle| = 1.3%

|%Error Circle| = 2.0%

Conclusion: 

Based on how small our percent errors were, I can conclude that the predicted periods I got through my derivations for the period of our two pendulums were very reasonable. However, there were basic assumptions we made that certainly contributed to our uncertainty, no matter how small they ultimately turned out to be.

Aside from the usual: possible inaccuracies in logger pro, uncertainty in our measuring instruments, etc, I observed two that were most significant to this lab:

First, when attaching the object through the hoops (made via the loops in a paperclip), we assumed that the pivot was perfect - without friction, attached perfectly about a pivot. However, this was not true, as the contact between the paper clip and the perpendicular hinge introduces friction about this point, which would lead to our period increasing over time (T = 2t, more time it takes to make a full swing results in a longer period). It's safe to assume this is a reasonable observation, as our experimental data is indeed larger than our predicted.

Finally, the cutout shapes weren't perfect by any means: the dimensions themselves were not perfect, with margins of error up to 1 cm. The assumption that we made by deriving a formula for inertia for a perfect triangle and semicircle would no doubt mean the actual inertia of the cutouts (whatever that may be) would not be exactly the same, albeit they still would be very similar.

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.