Feynman, in introducing his course at Caltech on gravitation, once recommended where to place our feet and heads. It was easy in studying gravity, he said, to have one's head in the clouds. In this course, he continued, we will keep our feet on the ground. Einstein did manage to do both. However, Feynman concluded, “Einstein was a very tall man!”

Lacking Einstein's height, I keep my feet on the ground through back-of-the-envelope reasoning. To illustrate this connection, I have chosen a problem between our and our students' level: the period of moderate-amplitude pendulum oscillations.1 

Since Galileo, the pendulum has starred in the history of mechanics.2 When we introduce students to it, we keep our feet on the ground by analyzing the pendulum using forces and torques, but we assume an infinitesimal amplitude. Advanced students learning mathematical methods revisit the pendulum free of the assumption. Then, the exact...

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