Question

The Achilles tendon is the strongest tendon in the human body. On average, the Achilles tendon of an adult is 150 mm in length and 20 mm2 in cross section. Surprisingly, it is quite elastic; it can store and release energy like a spring. But it's a very stiff spring, with a spring constant of approximately 2.5×106N/m. The force on the Achilles tendon while running can be 7 times the runner's weight.By how many mm does a 65 kg runner's Achilles tendon stretch?

Answer 1

To solve this problem, we have to use Hooke's Law.

$F=k\cdot\Delta x$

Where k is the constant of the spring, and x is the displacement we have to find.

According to the problem, the force is 7 times the runner's weight, which means

$F=7\cdot mg$

Where m = 65 kg and g = 9.8 m/s^2. Let's find this force.

$\begin{gathered} F=7\cdot65\operatorname{kg}\cdot9.8\cdot\frac{m}{s^2} \\ F=4459N \end{gathered}$

Then, we use Hooke's Law to find the displacement.

$\begin{gathered} 4459N=2.5\times10^6\cdot\frac{N}{m}\cdot\Delta x \\ \Delta x=\frac{4459N}{2.5\times10^6\cdot\frac{N}{m}} \\ \Delta x=1783.6\times10^{-6}m \\ \Delta x=1.7838\times10^{-3}m \end{gathered}$

But, we have to find the displacement in millimeters. So, let's divide by 1000.

$\begin{gathered} \Delta x=\frac{1.7838\times10^{-3}}{1000}mm \\ \Delta x\approx0.002\operatorname{mm} \end{gathered}$

Therefore, the runner's Achilles tendon will stretch 0.002 mm.