Question

The Achilles tendon connects the muscles in your calf to the back of your foot. When you are sprinting, your Achilles ten-don alternately stretches, as you bring your weight down onto your forward foot, and contracts to push you off the ground. A 70 kg runner has an Achilles tendon that is 15 cm long with a typical 1.1 * 10-4 m2 area. a. By how much will the runner’s Achilles tendon stretch if the maximum force on it is 8.0 times his weight, a typical value while running? b. What fraction of the tendon’s length does this correspond to?

Answer 1

A) We know that the gravitational Weight of a Body is given by,

$F=8mg$

The expression of young's modulus is given by,

$Y=\frac{Fl}{A\Delta l}$

Replacing the value of the force in the equation of young's modulis we have,

$Y = \frac{8mg*l}{A\Delta l}$

Re-arrange the equation for the rate of l, we have,

$\Delta l = \frac{8mgl}{AY}$

Our values here are giving by,

$m=70kg$

$g=9.8m/s^2$

$l=15*10^{-2}m$

$A=110*10^6m^2$

$Y = 1.5*10^{10}N/m^2$

Replacing in the previous equation we have that,

$\Delta l = \frac{8(70)(9.8)(15*10^{-2})}{110*10^6*1.5*{10}}$

$\Delta l = 4.98*10^{-3}m$

We note that Achilles tendon stretch around 5mm

B) The strain is given by the equation,

$\epsilon = \frac{\Delta l}{l}$

The fraction of the tendon's lenght is the ratio of the change in lenght to the streched lenght, that is

$\alpha = \frac{\Delta l}{l}$

$\alpha = \frac{5*10^{-3}}{15*10^{-2}}$

$\alpha = 0.033$

The fraction of the tendon's lenght is 0.0333