1. Series Expansion.
2. Elliptic Integral.
3. Period of Pendulum.

Estimated Time:50 minutes

The students

1. derive the expression for the period of the pendulum.
2. evaluate the full integral numerically and find the relevant parameters in the prefactor for the system to check absolute values.
3. plot the experimental result and the numerical results on the same graph.

For the series expansion of the integral, a thorough grounding in series expansion, as learned in “Symmetries and Idealizations”. For the computer evaluation of the integral, some knowledge of a package like Maple or Mathematica. The integral for the evaluation of the period of an oscillator, \(T=4\int\limits_{0}^{A}{\frac{dx}{\sqrt{\frac{2}{m}\left[ E-U\left( x \right) \right]}}}\) , should have been evaluated carefully in lecture for the case of a linear harmonic oscillator, (e.g. a mass on a spring, and the symbols have their usual meanings) \(U\left( x \right)=\frac{1}{2}kx^{2}\) . The independence from the amplitude will have been discussed.

• Computers with Maple
• A handout for each student. Here's a Word document version handout.

Students work in groups. Three is maximal; for this activity, two may be better.

Provide overall guidance about the goal and direction, with assurances of mini wrap-ups along the way.

1. The first section involves the mapping from linear displacement ($x$) to angular displacement $(\theta)$. Encourage the students to recall the appropriate angular quantities from their introductory work.
2. The second section involves cleaning up the integral and pulling all dimensional terms outside. Discuss with the students the desirability of this approach and set them to the task, but warn them not to try to evaluate any integrals at this stage.
3. The third section requires either (a) evaluation of the integral by computer, or (b) substitution of the series expansion for \(\cos \theta \) AND \(\cos \theta _{\max }\), and gathering of like power terms.
4. The fourth section is to evaluate the (simpler) integrals, which are similar to the one already demonstrated for the SHO.

1. The goal of the first section is to obtain $$T=4\int\limits_{0}^{\theta _{\max }}{\frac{d\theta }{\sqrt{\frac{2}{I}\left[ MgL\left( 1-\cos \theta _{\max } \right)-MgL\left( 1-\cos \theta \right) \right]}}}$$. Students usually remember that they should use the moment of inertia, but often think they need the actual form for a rectangular rod, when it is desirable to leave the quantity as “$I$”. They invariably use the total length of the pendulum, but someone will sometimes remember that the distance to the center of mass should be used (L in the above equation). Most students are used to the force method to analyze simple harmonic motion, and are not immediately able to write down the potential energy. Encourage them to draw a picture. Identifying the total energy E with the potential energy at maximum angular displacement is also not obvious, until explicit questioning refers to the same process in the lecture example.
2. The goal of the second section is to obtain $$T=\sqrt{\frac{2I}{MgL}}\int\limits_{0}^{\theta _{\max }}{\frac{d\theta }{\sqrt{\cos \theta _{\max }-\cos \theta }}}=T_{\text{small angle}}\frac{\sqrt{2}}{\pi }\int\limits_{0}^{\theta _{\max }}{\frac{d\theta }{\sqrt{\cos \theta _{\max }-\cos \theta }}}$$. Students usually see the MgL cancellation, and usually make algebra mistakes while clearing factors. Some remember to use dimensional analysis to help troubleshoot, but this is a good time to encourage that. If the class is experienced, identifying the small angle period at this point is in order, but if they're struggling, leave it to the end.
3. The goal of the third section is to obtain the integral by using a computer package. Some manipulation of the integral is necessary to force the expression into the standard form for an elliptic integral of the first kind. The students have not seen this particular integral before we simply take the view that it is a tool and they don't need to know much about it. We tell them the substitution to use to manipulate the expression: \(t=\frac{\sin \left( \frac{\theta}{2} \right)}{\sin \left( \frac{\theta _{\max }}{2} \right)}\). Students are usually not yet familiar with the idea that a “constant” in an integral (in this case \(\cos \theta _{\max }\)), can be a variable parameter and therefore the integral can be evaluated for many such values. There is usually discussion about what values to substitute into the integral, and the parameter discussion comes up naturally.

Here is a Maple worksheet that evaluates the elliptic integral.

• The goal of the fourth section is to evaluate the (simpler) integrals above, which are similar to the one already demonstrated for the SHO. A few students recognize that the first term is the “easy” integral evaluated for the simple harmonic example. What is new to them is that there is another term, the same integral multiplied by \(\frac{\theta _{\max }^{2}}{24}\) that represents a correction dependent on the amplitude. Even more surprising to them is that the final term contains the same \(\theta _{\max }^{2}\) term, but this time it comes from the integral limit. The final result is

\[T=T_{\text{small angle}}\left( 1+\frac{\theta _{max}^{2}}{16} \right)\]

Mini wrap-ups are in order at several steps in the calculation, so the students don't get lost (and they do, very easily)

• Wrap up the first two sections by ensuring that students ask all the questions they need to about getting to the \[T\left( \theta _{\max } \right)\] expression, and cleaning it up.

• Wrap up the third section with a discussion of constants/parameters, whether the model and experiment are in quantitative agreement. Ask the students to plot explicitly the period/amplitude relationship, including a plot of the prediction of a purely harmonic oscillator model. This concludes the activity for this approach.

• Wrap up the fourth section with whether the model and experiment are in quantitative agreement. Ask the students to plot explicitly the period/amplitude relationship, including a plot of the prediction of a purely harmonic oscillator model. The discussion here will surely include whether it was “worth it” to get a result that “doesn't agree” (the students' words). The instructor should have on hand a plot of the period/amplitude relationship for the SHO model, the refinements with the next two terms in the cosine expansion, and the exact result from the evaluation of the elliptic integral.

Calculation of Pendulum Period (Power Series) is an alternate way to do this activity. Instead of evaluate the integral using the numerical approach, the alternate version of this activity used the series expansion.