The following equation can be used to calculate muscle fiber force () at an activation (), fiber length (), and fiber velocity ():

The functions  and  represent the normalized active force–length, force–velocity, and passive force–length curves, respectively; and represents the maximum isometric muscle force. How does this equation simplify in each of the following cases?

Muscles fibers are:

(a) isometric

(b) at optimal fiber length

(c) maximally activated

(d) passive

Moment arms determine the moment-generating potential as well as the amount a muscle changes length with joint rotation. Moment arms vary with joint angle. They are determined by the geometry of the joint: center of rotation, and the relative locations of the attachment points of the muscle with respect to the joint. This question allows you to explore this relationship:

(a) Two important parameters about the moment arm vs. joint angle relationship are: the peak moment arm and the angle at which the moment arm peaks. In order to get a larger peak moment arm, would you want your muscle attachments to be closer to or further from the joint?

(b) Based on the triangle model below, determine the value of the peak moment arm () and the angle () at which the moment arm peaks, in terms of the distances from the attachment sites to the joint center,  and  . Assume that  < .

(c) What are the limitations to assuming a model such as this one?

(d) Why are peak moment arm and the angle at which the moment arm peaks important?

Muscles Hans and Franz have both entered the prestigious Muscle Olympics (leaving their pesky tendons at home). They will each compete in three events in an attempt to bring home the coveted Golden Sarcomere. Being the poor student that you are, you’re hoping to make a little extra cash by placing some bets on the competition. The Olympic program with the competitor’s vital stats has just arrived in the mail and your bookie is on the phone – what bets do you want to place?

(a) For each event, predict whether Hans or Franz will win. Defend your choice using muscle mechanics principles and equations we’ve discussed in class. (You get credit for your reasons, not for just correctly guessing the winner). If there is a muscle property that you think is important that isn’t listed, assume that it is the same for both Hans and Franz. Assume both muscles have linear force-length properties and moment arms that are constant with joint angle. For all events, the muscles are loaded into a revolute joint with the given moment arm.

“Clean and Jerk”: The joint moves through a range of motion at a constant angular velocity of 100°/sec. The winning muscle is the one that produces the most force at the instant that it reaches optimal fiber length.

“60° Torque-off”: Each joint is fixed at 60°. The winning muscle is the one that produces the largest torque during an isometric contraction. Assume that each muscle is at optimal fiber length when the joint is at 60°.

“110° Torque-off”: Each joint is fixed at 110°. The winning muscle is the one that produces the largest torque during an isometric contraction. Assume again that each muscle is at optimal fiber length when the joint is at 60°.

(b) Hans and Franz were not both satisfied with their performance in the Torque-off competitions—they each thought they should be able to win both the 60° and 110° categories. They convince the Olympic officials to let them have a rematch. Over the weekend, both muscles worked hard to try to improve their odds. Franz went to the local Gold’s Gym and bulked up. Hans went to the local yoga studio and stayed in downward-facing dog all weekend. After this intense conditioning, their vital stats had changed, but each muscle still reaches its optimal fiber length at 60°—get your bookie back on the phone and predict who will win the two Torque-off competitions now.

(c) Using what you learned in this problem, comment on why we may have redundant muscles in our bodies.

Disclaimer: Don’t take Hans’s and Franz’s experiences literally. Only a chronic stretch and a sustained weight training program could actually change the properties of your muscles – but you already knew that.

A young biomechanist, Ted, is attempting to estimate a muscle’s active force–length curve. However, he cannot isolate the muscle of interest, but instead can only perform experiments on the muscle–tendon unit as whole. He performs a series of trials where the muscle–tendon unit is held isometric and tendon force is measured in steady state. First, he plots the passive (a(t) = 0) force curve. Later, he measures the fully activated (a(t) = 1) force curve, which includes passive muscle force (we have called this the total force curve). By subtracting the passive curve from the fully activated curve, he hopes to accurately estimate the muscle’s active force–length curve.

We will compare Ted’s estimated active force–length curves with the active force–length curve of muscle alone in two conditions:

• stiff tendon (tendon slack length is 1/10 as long as optimal fiber length)
• compliant tendon (tendon slack length is 10 times the optimal fiber length)

Assume the muscle and tendon properties are linear.

(a) To make this comparison, we will plot force–length curves using the axes below. Explain what the x- and y-axis units represent.

(b) Plot the passive force–length curve for both the stiff and compliant tendon on the axis above to make a comparison between the two.

(c) Plot the total force–length curve for each muscle–tendon unit (one with the stiff tendon, the other compliant) on the same axis to make a comparison between the two.

(d) Find the active force–length curve for each muscle–tendon unit by subtracting the passive force–length curve from the total force–length curve.

(e) Discuss your results. Is Ted’s method reliable? Why or why not?

One way to estimate moment arm of a muscle with respect to a particular degree of freedom is to measure the length of the muscle–tendon complex over a range of joint angles and then use the following equation:

where  is the moment arm, is the muscle–tendon length, and  is the joint angle. Prove the above equation. You may prove this with geometry or the principle of virtual work.

Given the neural excitation, , shown below, derive an expression for and plot the muscle activation, , as a function of time. Label any critical points in your graph. Assume that the time constant for activation is 0.12 seconds and the time constant for deactivation is 0.24 seconds.

Feel free to use either the excitation–activation dynamics ODE from your textbook or the ODE shown below to do your calculations.

Excitation–activation dynamics ODE:

where