Chapter 11, Problem 6Q

S. What is the approximate period of your walking step?

To determine

The approximate period of walking step.

Answer to Problem 6Q

Solution:

  $T=2\pi \sqrt{\frac{2I}{3g}}$

Explanation of Solution

While walking, one leg swings ahead while the other stays on the ground. The leg which swings ahead uses muscle force and then relaxes. The leg which stays on the ground stays straight, while the leg which swings forward, does it so by bending the knee slightly which allows it to pass below the body.

The leg which swings forward can be modeled as a physical pendulum. It can be considered as a thin uniform rod of mass m which rotates about a point at a distance r from the center of mass. Considering the leg to be swing freely under gravity g. The moment of inertia of the leg be I and the period of swing leg be T:

  $T=2\pi \sqrt{\frac{I}{mgr}}$

For a uniform, thin rod of length L with a pivoted end, $I=\frac{m{L}^2}{3}$

The natural walking step length is roughly $\frac{L}{5}$ . The natural speed of walking v is the step length divided by the time required to take the step.

To move faster or slower than the natural speed, the legs do not move at their natural frequencies or with the natural step length. Instead, the muscles exert a force (hence torque) to move the body forward. The maximum force a muscle can produce $F_{Max}$ is proportional to its cross-sectional area A, which is proportional to the square of the length L

  $F_{Max}\propto A\propto L^2$

The maximum torque $T_{Max}$that the muscle can exert about its pivot point is proportional to the product of $F_{Max}$ and its length L

  $T_{Max}\propto F_{Max}L$

The mass of the leg is proportional to the mass of the muscle. Unlike the case of walking, the period of a pendulum acted on by maximum torque $T_{Max}$ is (constant of proportionality not organism-specific):

  $T\propto \sqrt{\frac{I}{T_{Max}}}$

The natural period with which the leg swings is

  $T=2\pi \sqrt{\frac{2I}{3g}}$

Where, $g=9.8{\text{m/s}}^{\text{2}}$ is the acceleration due to gravity and the period is independent of walking speed.

Conclusion: The approximate period of our walking step is $T=2\pi \sqrt{\frac{2I}{3g}}$ .