Chapter 15, Problem 36AP

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 ℓ, swinging as a physical pendulum through one-half of a cycle, in resonance. Let θ_max represent its amplitude. (a) Show that the animal’s speed is given by the expression

$v=\frac{\sqrt{6g\mathcal{l}}\sin {\theta }_{\max }}{\pi }$

if θ_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

$v=\frac{\sqrt{6g\mathcal{l}\cos ({\theta }_{\max }/2)}\sin {\theta }_{\max }}{\pi }$

(b) Evaluate the walking speed of a human with leg length 0.850 m and leg-swing amplitude 28.0°. (c) What leg length would give twice the speed for the same angular amplitude?