Some Examples of Static Equilibrium — Question 7

Question 7

A short tendon moment arm magnifies internal forces

In a simplified static foot model, the ankle is at $x=0$, an upward tendon force acts at $x=-.050\,\mathrm m$, and an upward toe-ground force of $600\,\mathrm N$ acts at $x=.150\,\mathrm m$. Neglect foot weight; the ankle force acts at $x=0$.

1. Find tendon force.

2. Find the ankle force on the foot and explain why it exceeds the ground force.

Question 7 - Solution

A short tendon moment arm magnifies internal forces

(a) Derivation and calculation

Take moments about the ankle so its unknown force has zero lever arm: $$-.050T+.150(600)=0.$$ Thus $$\boxed{T=1800\ \mathrm N\ \text{up}}.$$

(b) Result and interpretation

Let $A_y$ be the ankle force on the foot, positive upward. Vertical equilibrium gives $$A_y+T+600=0,$$ $$\boxed{A_y=-2400\ \mathrm N},$$ so the ankle force on the foot is 2400 N downward. The opposing contact force on the leg is upward by Newton’s third law.

Physical check

The short tendon lever arm requires three times the toe force; its moment is $1800(.05)=90$ N m.