Question

When a sprinter is attempting to perform an all-out sprint as seen in the panel above, in order to maximize their linear velocity they will flex the knee more during the initial portion of the swing phase than the knee would naturally flex during this phase. This alters the moment of inertia of the leg in a way that ultimately leads to increased linear velocity of the foot at the instant before foot strike. Describe the steps in this process. 1) Define moment of inertia, and describe how it changes as the knee goes through exaggerated flexion during the swing phase. 2) Describe what effect this change has on the angular velocity of the leg about the hip through the swing phase. 3) How is the angular impulse altered about the hip, and what relationship does this have to the change in angular velocity?

Answer 1

1) The moment of inertia is a measure of resistance to rotation, just like the mass of an object is a measure of resistance to a translation motion. This is evidenced in Newton's 2nd Law for rotation:

Which is analog to F = ma. The moment of inertia of a punctual mass about a given an axis of rotation depends on the radius (perpendicular distance from axis to mass) and the mass magnitude itself

We can generalize this definition to systems of particles and furthermore continuous bodies using the principle of superposition:

When he knee goes through exaggerated flexion during the swing phase, we are increasing the radius of rotation, and therefore increasing the total moment of inertia about the principal axis (axis passing through the COM)

2) The effect of increasing the moment of inertia is a decrease in the angular velocity of the leg, this must be so for angular momentum to be conserved.

3) Subsequently as the motion progresses the inertia is decreased, and the angular velocity is therefore increased, to mantain the angular momentum constant . This leads to increase the linear velocity of the foot before the strike.