Serway & Jewett — Oscillatory Motion: Problem 36

To account for the walking speed of a bipedal or quadrupedal animal, model a leg that is not contacting the ground as a uniform rod of length \(\ell\), swinging as a physical pendulum through one-half of a cycle, in resonance. Let \(\theta_{\text{max}}\) represent its amplitude. (a) Show that the animal’s speed is given by the expression \[v = \frac{\sqrt{6g\ell} \sin \theta_{\text{max}}}{\pi}\] if \(\theta_{\text{max}}\) is sufficiently small that the motion is nearly simple harmonic. An empirical relationship that is based on the same model and applies over a wider range of angles is \[v = \frac{\sqrt{6g\ell \cos(\theta_{\text{max}}/2)} \sin \theta_{\text{max}}}{\pi}\] (b) Evaluate the walking speed of a human with leg length \(0.850 \text{ m}\) and leg-swing amplitude \(28.0^\circ\). (c) What leg length would give twice the speed for the same angular amplitude?