Question

Tendons are, essentially, elastic cords stretched between two fixed ends; as such, they can support standing waves. These resonances can be undesirable. The Achilles tendon connects the heel with a muscle in the calf. A woman has a 20-cm-long tendon with a cross-section area of 130 mm2 . The density of tendon tissue is 1100 kg/m3.

For a reasonable tension of 500 N , what will be the fundamental resonant frequency of her Achilles tendon?

Answer 1

The mass is,

$$ \begin{aligned} M &=\rho V \\ &=\rho A L \\ &=(1100)\left(130 \times 10^{-6}\right)\left(20 \times 10^{-2}\right) \\ &=0.0286 \mathrm{~kg} \end{aligned} $$

The linear density is,

$$ \begin{aligned} \mu &=\frac{M}{L} \\ &=\frac{0.0286}{20 \times 10^{-2}} \\ &=0.143 \end{aligned} $$

The fundamental resonant frequency is,

$$ f=\frac{1}{2 L} \sqrt{\frac{T}{\mu}} $$

$$ \begin{array}{l} =\frac{1}{2(0.2)} \sqrt{\frac{500}{0.143}} \\ =147.828 \mathrm{~Hz} \end{array} $$

\(=150 \mathrm{~Hz}(\) two significant figures \()\)