## Muscle Contractions

An animal muscle consists of a large number of approximately parallel fibers of approximately equal cross-sections. See Fig 5.1. In a muscle contraction, each of these fibers contracts while exerting approximately equal force. The total force F exerted by the muscle is therefore proportional to the number of contracting fibers and thus to the cross-sectional area A of the entire muscle: F∝ A.

A muscle of mass M contracting with speed V has a kinetic energy MV²/2. This energy is equal to the work done by the force F over the contraction distance D: MV²/2 = FD. The mass M is proportional to the product AD of the area A and distance D so that V² is proportional to FD/M ∝ AD/AD and is therefore independent of the size of the muscle. The speed V is therefore said to be scale invariant.

A consequence of this scale invariance is that all animals of similar shape (racing dogs and horses, cats and tigers, dolphins and whales) run (swim) at approximately the same maximum speeds. Furthermore, this is born out across the species spectrum: most mammals sprint at about 15 - 20 m/s. The outliers are cobras, who sprint the fastest at about 30 m/s, and humans, who sprint the slowest at about 10 m/s. (It helps to have four legs.) Of course, runners in a sprint do not all run at the same speed or finish at the same time. The best sprinters have beneficial physiology such as relatively large leg muscles and small body weight.

Running is a relatively inefficient process because, when the rear leg propels the runner’s body off the ground and the forward moving front leg strikes the ground, the front leg exerts a backward force on the body, which causes the body to temporarily slow down. This breaking effect is obviously inefficient, but as the body then rotates forward over the front foot, the Achilles tendon that attaches the heel to the calf muscles is stretched, storing elastic energy that would otherwise be dissipated as heat. See Fig. 5.2. When the now backward foot on the ground then rotates forward to propel the body upward and forward, the required muscle force is assisted by the release of the energy stored in the tendon. This elastic energy transfer greatly increases the efficiency of running.

Unlike running, walking is not scale invariant. (In walking, at least one foot is on the ground at all times. In running, both feet are off the ground for a part of time.) The most efficient walking speed occurs when the moving leg swings forward as a pendulum, powered mainly by gravity with minimal muscular contribution. This speed is w0 = √(2lg)/𝜋, where l is the leg length of the walker and g is the acceleration of gravity. Walking at a slower or a faster speed requires a greater muscular contribution and is therefore less efficient. The maximum possible walking speed w1 occurs when the circular acceleration v²/l is equal to g, so that w1 = √(lg). This speed is about 3 m/s for a typical adult male.

When a muscle contracts while exerting a force f > 0 (such as while lifting a weight of magnitude f), it’s maximum contraction speed v is less than the maximum contraction speed V obtainable when the muscle exerts zero force (f = 0). This maximum speed v decreases when f increases, and becomes 0 when the muscle exerts it’s maximum force f = F. The power p = fv exerted by the muscle is therefore 0 when v = V and f = 0 and also when v = 0 and f = F. In between these end values, p rises to a maximum value P of about FV/9 when f ≈F/3 and v ≈ V/3.