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

Estimated Time: 50 minutes

1. Students were asked to derive the expression for the period of the pendulum.
2. Students evaluated the expression using the series expansion.
3. Plot the experimental result and the numerical result 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.

• Tabletop Whiteboard with markers
• A handout for each student handout

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 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.

• 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.

• 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.

* The goal of the third section is substitution of the series expansion for \(\cos \theta\) AND \(\cos \theta _{\max }\). There is always discussion of how many terms to use in the expansion. Many students want to stop at the squared term, failing to recognize that this gives the harmonic approximation. Others want to include many terms. The interesting part is that students are reluctant to use the series approximation at all, pointing out that the oscillations were large-amplitude (close to 180$^{\circ}$ in some cases). Instructors guide the discussion towards the conclusion that a series approximation gives the correct results for all angles if enough terms are included, and the only question is how much is “enough”. Students struggle with the idea that there is no hard-and-fast rule for “enough”, and that it depends on experimental accuracy amongst other things. Once they plot the results, though, the concept of “enough terms” begins to make sense. (The squared term only is not enough - it predicts a constant amplitude, clearly inconsistent. The quartic term produces consistency to about 45$^{\circ}$ - better). Gathering of like powers and factoring out the common factor of \(\left( \theta _{\max }^{2}-\theta ^{2} \right)\) is difficult for students, as is recognizing that the “downstairs” series expansion must be moved “upstairs”. This is often instructor-demonstrated. At the end of this section, the expression should look like this: \[T=\sqrt{\frac{2I}{MgL}}\int\limits_{0}^{\theta _{\max }}{\frac{d\theta }{\sqrt{\left( \theta _{\max }^{2}-\theta ^{2} \right)}}\left( 1+\frac{1}{24}\left( \theta _{\max }^{2}+\theta ^{2} \right) \right)}\]

• 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, and where the calculation will go next.

• 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 (Numerical Approach) is an alternate way to do this activity. Instead of evaluating the integral using the series expansion, the alternate version of this activity uses a numerical approach. This numerical approach is better when this course precedes the “Symmetries” course. “Symmetries” introduces power series expansions, and if this basis is not present, then the power series expansion is too difficult.