## Related questions with answers

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 $\theta_{\max }$ represent its amplitude. What leg length would give twice the speed for the same angular amplitude?

Solution

Verified$\textbf{Calculation: }$ Solve for first case: From part (a), the pendulum speed is given by

$\begin{align*} v_{1} &= \dfrac{ \sqrt{ 6 g L' ~ \cos \left( \dfrac{ \theta_{max} }{2} \right) } ~ \sin \left( \theta \right) }{ \pi } \\ \end{align*}$

Solve for second case: From part (b), the pendulum speed is given by

$\begin{align*} v_{2} &= \dfrac{ \sqrt{ 6 g L ~ \cos \left( \dfrac{ \theta_{max} }{2} \right) } ~ \sin \left( \theta \right) }{ \pi } \\ \end{align*}$

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