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.