Question

Tendons are, essentially, elastic cords stretched between two fixed ends. As such, they can support standing waves. A woman has a 20-cm-long Achilles tendon—connecting the heel to a muscle in the calf—with a cross-section area of 90 mm^2.. The density of tendon tissue is 1100 kg/m³. For a reasonable tension of 500 N, what will be the fundamental frequency of her Achilles tendon?

Short answer

The fundamental frequency of the tendon is 177.5 Hz

Step-by-step solution

Write the given information 

The length of the tendon, L= 20 cm = 0.2m
Area of cross section, A= 90 mm² = 90x 10^-6 m²
The density of the tendon, ρ= 1100 kg/ m³
The tension in the string, T= 500N

To determine the fundamental frequency of tendon 

The mass per unit length of the string is given by
$$\begin{gathered} \mu =\rho A \\ =(1100kg/m^3)(90x{10}^{-6}{m}^2) \\ =0.099kg/m \end{gathered}$$

The frequency of the tendon is given by

$$\begin{gathered} f=\left(\frac{1}{2L}\right)\sqrt{\frac{T}{\mu }} \\ =\left(\frac{1}{2(0.2)}\right)\sqrt{\frac{500N}{0.099kg/m}} \\ =2.5\left(71\right) \\ =177.5Hz \end{gathered}$$

Thus, the fundamental frequency of the tendon is 177.5 Hz