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This problem was contributed by Jon Stingel at Stanford University.

Suppose a muscle–tendon unit has the following properties:

• Mathinline
  body --uriencoded--\ell_o%5eM = 30~\text%7Bcm%7D

• Mathinline
  body --uriencoded--F_o%5eM = 1000~\text%7BN%7D

• Mathinline
  body --uriencoded--\ell_s%5eT = 10~\text%7Bcm%7D

and it is exerting force between two bones in the following scenario:

• Mathinline
  body --uriencoded--\ell%5e%7BMT%7D = 40~\text%7Bcm%7D

• Mathinline
  body a=1

• pennation angle

  Mathinline
  body \phi = 0

(a) Assume the tendon is rigid, the muscle is contracting isometrically, and the muscle and tendon are in equilibrium. Calculate the length of the muscle (

Mathinline
body --uriencoded--\ell%5eM

) and the force in the muscle (

Mathinline
body --uriencoded--F%5eM

).

(b) Now assume that the tendon is compliant with stiffness

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body --uriencoded--k = 100~\text%7BN/cm%7D

. Further assume that both the muscle and tendon have linear force–length relationships (see figures below). Design an algorithm (using pseudocode or a flowchart) to solve for the muscle force (

Mathinline
body --uriencoded--F%5eM

), starting from an initial guess for muscle length (

Mathinline
body --uriencoded--\ell%5eM

) and iterating until a stopping condition is met. Just describe the algorithm (do not try to solve for

Mathinline
body --uriencoded--F%5eM

).

(c) Complete one iteration of the algorithm you designed in part (b) using the properties and scenario described above. Use the initial guess

Mathinline
body --uriencoded--\ell%5eM = 22.5~\text%7Bcm%7D

. Based on your calculations, how should you change your guess to get closer to the point where the muscle and tendon are in equilibrium?

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Include Page BMH:S5.6 Muscle–tendon equilibrium: rigid vs. compliant tendon BMH:S5.6 Muscle–tendon equilibrium: rigid vs. compliant tendon