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--long tendon with a cross-section area of 130 . The density of tendon tissue is 1100 .

For a reasonable tension of 600 , what will be the fundamental resonant frequency of her Achilles tendon?

Answer 1

161.9 Hz ( fundamental resonant frequency )

other resonant frequencies: 323.85 Hz, 485.7 Hz..

Step-by-step explanation:

Given:

A= 130
`mm^(2)` => 130 x
`10^(-6)`
`m^(2)`

Tension T= 600N

Length L= 20cm= 0.2m

Density of tendon ρ= 1100 kg/
`m^(3)`

Linear mass density is defines as:

μ = m/L = ρV/ L = ρAL / L

μ = ρA

where,

m=mass , V = volume, L= length , A= cross section area and ρ= density

so, μ = 1100 x 130 x
`10^(-6)` => 0.143 kg/m

Wave speed in the string is defines as

v= sqrt(T/μ)

where,

T is string tension and μ is the linear mass density.

So,

v=
`\sqrt{(600)/(0.143) }`

v= 64.77 m/s

Frequencies of standing wave- modes of a string of length L fixed at both ends can be defines as:

fm = m (
`(v)/(2L)` ) where m= 1,2,3,4,.....

Therefore, fundamental resonant frequency of her Achilles tendon is:

`f_(1)` =
`(64.77)/(2 * 0.2)` => 161.9 Hz

The other resonant frequencies can be find by integral multiples of frequence.

So,

`f_(n) = n * f_(1)`

`f_(2) = 2 * 161.9` = 323.85 Hz

`f_(3) = 3 * 161.9` = 485.7 Hz