Lab 22: Physical Pendulum

We have previously worked with ideal pendulums, theoretically mass-less besides a point mass at the end. In reality, pendulum motion occurs across many different kinds of objects of different configurations. Today, we solve for the period of a semi-circle attached at the top and bottom, and swung sideways.

She even has a sexy mole. *Ahem* Anyways, here's an illustration of the different points on the circle. The radius of the circle is 14.6 cm.

To find period, we recall that finding the period before requires finding the angular velocity, which is found by acceleration (in this case, angular acceleration). One way of finding some relationship relating acceleration in this case is:

Therefore, we see that we need to find the moment of inertia and the center of mass. If we imagine it a whole circle in which the pivot is in the center, then this half circle pivoted at the top is just the circle with half the mass, so it is possible to infer that the moment of inertia at that point is just:

To find the center of mass, we must get our hands dirty with some calculus:

Now, we have what we need to analytically solve for the period, with the pivot on top. We also simplify our calculations by assuming small angles, replacing sinθ with θ.

Using a photogate, as pictured above, and Logger Pro, we experimentally find the period with similar results:

Ignore the large box on the bottom, which is inaccurate because it shows the period after running for longer than it should. We can look at the data in the table instead, and see that the period is roughly 0.86s before the effects of friction, among other things kicked in.

We can also turn the semi-circle around and pivot it at the bottom:

Analytically, this is a mess. We must first use the parallel axis theorem to find the inertia at the center of mass using our inertia at the top and then use this new center of mass inertia, and parallel axis again to find the inertia at the bottom:

Whew! With that done, we use the torque equation, as before, to solve for the period with the pivot at the bottom of the circle:

And experimentally:

Once again, we ignore the period box on the bottom and see that the period is actually around 0.83s. An overview of results is in order:

	Results		Error
	Analysis (s)	Experiment (s)	Absolute (s)	Relative (%)
Top	0.832	0.86	0.028	3.40%
Bottom	0.816	0.83	0.014	1.70%

Pretty close, I should say! And to round out the semester, here's a Dendrobium goldschmidtianum from my collection that just bloomed:

Thanks for the semester, Professor Wolf. It was hard, but you were reasonable, answered our questions when we had them, and even taught us a little about local Walnut politics.