If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Physics of walking and running

## Problem

Walking is energy efficient. In a walking human, one leg swings forward while the other leg’s foot stays planted on the ground. When walking at natural speed (defined below), the swinging leg uses muscle force to move forward and immediately relaxes, allowing the force of gravity to move it to the ground. Simultaneously, the planted leg moves forward with largely passive rotation at the hip. The plant leg only needs to stay straight and the swinging leg’s knee only slightly bends to allow it to pass underneath the body.

Figure 1: Leg positioning. Plant leg in blue, swinging leg in red.
The swinging leg can be modeled as a physical pendulum: a thin uniform rod of mass $m$ rotating about a point a distance $r$ from its center of mass. Swinging freely under gravitational acceleration, $g$, such a physical pendulum with moment of inertia $I$ will swing with a period $T$:
$T=2\pi \sqrt{\frac{I}{mgr}}$ (Equation 1)
For a uniform, thin rod of length l with a pivoted end, $I=\frac{{ml}^{2}}{3}$. 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 produce forces (hence torques) 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: ${F}_{max}\propto A\propto {l}^{2}$. The maximum torque that the muscle can exert about its pivot point, ${L}_{max}$, is proportional to the product of ${F}_{max}$ and its length: ${L}_{max}\propto {F}_{max}l$. The mass of the leg is proportional to the mass of the muscle, which is proportional to the product of the area and the length. Unlike in the case of walking, the period of a pendulum acted on by maximum torque ${L}_{max}$ is (constant of proportionality not organism-specific)
$T\propto \sqrt{\frac{I}{{L}_{max}}}$ (Equation 2)
Modeling the leg as a uniform cylindrical rod, which of the following changes would most increase the moment of inertia of a runner’s leg?