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# 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, equals, 2, π, square root of, start fraction, I, divided by, m, g, r, end fraction, end square root (Equation 1)
For a uniform, thin rod of length l with a pivoted end, I, equals, start fraction, m, l, squared, divided by, 3, end fraction. The natural walking step length is roughly start fraction, l, divided by, 5, end fraction. 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, start subscript, m, a, x, end subscript, is proportional to its cross sectional area, A, which is proportional to the square of the length: F, start subscript, m, a, x, end subscript, ∝, A, ∝, l, squared. The maximum torque that the muscle can exert about its pivot point, L, start subscript, m, a, x, end subscript, is proportional to the product of F, start subscript, m, a, x, end subscript and its length: L, start subscript, m, a, x, end subscript, ∝, F, start subscript, m, a, x, end subscript, 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, start subscript, m, a, x, end subscript is (constant of proportionality not organism-specific)
T, ∝, square root of, start fraction, I, divided by, L, start subscript, m, a, x, end subscript, end fraction, end square root (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?
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