Musculoskeletal systemThis chapter analyzes the gross aspects of theskeleton, muscles and tendons as a system of ar-ticulated levers operated by ropes, pulleys andcontractile units. The object of this system is toexert forces of various kinds on the externalworld—for example to move the animal (locomo-tion) or to procure food.1. Skeletons of land verterbratesThe verterbrate skeleton consists of bone1, a com-posite material consisting of bone cells, collagen(a fibrous protein arranged in long strands), andinorganic rod-like crystals of Ca10(PO4)6(OH)2,perhaps 50 Å in diameter and 200 to 2,000 Å long.The quantitative properties of bone are given inthe Table below:Properties of Bone Type of StressUltimate Strength(× 108 Nt ⁄ m2)Compression 1.5Tension 1.2 — 1.5Bending 2.1 — 2.2Young’s modulus 171 — 1852. MuscleMuscle tissue consists of basic contractile unitscalled sarcomeres. The sarcomeres are attachedend to end, with the demarcations marked byZ-membranes. At the microscopic level, the sar-comere contracts because the cross-bridges on themyosin fibers ratchet along the actin fibers. Wediscuss this in more detail in Chapter X, Bioener-getics.The sarcomere can move only one way: it con-tracts. Thus (contractile) muscular force must bebalanced by a weight, a spring, or an opposingmuscle, in order that the fibers can be pulled backto their initial uncontracted state.The important things to remember about musclesare 1. they can exert a maximum stress of 3×105Nt/m2; and 2. they can contract at most 20–25% of theiroverall length (maximum strain=0.2–0.25).Physics of the Human Body 23Chapter 3: Musculoskeletal system1. Data on the properties of bone were taken from the article: “bone”, Encyclopædia Britannica Online(Copyright © 1994-2001 Encyclopædia Britannica, Inc.).3. Mechanical (dis)advantageWe now analyze the muscular contraction forceneeded to lift a given weight. To do this we musttake into account the articulated bones, which actsomewhat like scissors jacks. Because muscle can contact at most 25% of itslength, the arrangement of vertebrate skeletal lev-ers actuated by muscular contraction sacrificesmechanical advantage for range of motion. A sec-ondary result is that the “output” achieves absolutespeeds much greater than those of the muscularcontractions. A second constraint on the evolutionary optimiza-tion of organisms is the fact that muscles can onlyexert force while contracting. To make a musclereturn to its uncontracted state it is necessary topair it with a muscle that, when contracting,stretches the opposing muscle2. That is, all limbicmuscles occur in pairs, called flexors and extensors.Virtual workTo analyze the levers and muscles comprising alimb, we apply the principle of virtual work3. Forexample, suppose the leg shown to the right raisesthe weight W by a distance δh, and thereby per-forms work W δh. In so doing, the angle of thefemur (thighbone) from the vertical changes fromθ to θ − δθ. Taking the femur and tibia (shinbone) to be thesame length L, and the muscles and tendons tohave the (greater) length l, we see from the (Py-thagorean theorem) thatl2 = L2 + 2Ldsinθ + d2where d is the offset of the muscle attachments.The muscle exerts a force T over the distance δlhence does work T δ l. Since the tendon attachesbelow the knee, we setW δh ≈ 2T δ land noting that δh = −2Lsinθ δθ ,at last findT = W tanθ 1 + 2Γsinθ + Γ21⁄2where Γ is the ratio L⁄d . For small angles from thevertical, very little tension is needed to raise or24 Chapter 3: Mechanical (dis)advantage2. This is not universal—exoskeletal animals like spiders extend their muscles using a hydraulic system,whereas certain other muscles compress springy tissues that expand when the muscles relax.3. See, e.g., H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley Publishing Co., Reading, MA,1980), p. 17ff.lower the weight. However, for angles beyond 45othe force rises very rapidly indeed. For my leg theratio Γ is about 3.3, hence at an angle of 45o themechanical (dis)advantage (at 45o) is about 4.Thus the tension must equal 4 times the weight tobe lifted. But at an angle of 60o the tension rises to7 times the weight. This is why exercise physiolo-gists warn us not to perform knee bends at anglesexceeding 45o—we can damage our knee liga-ments at higher angles!ExampleLet us estimate how much I can bench press. Myupper arm has a circumference of 15 inches or38 cm. Setting this to 2πr I find the radius of myupper arm muscles (biceps and triceps) to ber ≈ 6 cm. Their total cross-section is thus some114 cm2. The cross-section of the upper arm bone(humerus) is perhaps 3 cm2, so we may take thearea of the triceps to be about 74 cm2 (about 2⁄3the difference). Multiply by the maximum stress,3×105 Nt/m2, to get a net force of 2220 Nt per armmuscle. From the preceding analysis we see thatthe net weight I can lift (using my triceps alone) isW = Tmax d l cotθ ;with L=32 cm and d=7 cm, and with θ=45o, weget W ≈ 2 × 22205.3 Nt ≈ 189 lb .This is very close to the amount I can currentlybench press (10 repetitions), about 180 lb.4. Hill’s LawMany of the ideas in this section are taken fromthe excellent book by C.J. Pennycuick.4 The physi-ologist A.V. Hill proposed the following empiricalrelationship5 betweenthe absolute speed of muscu-lar contraction and the force being exerted by themuscle:v = v0 Fmax − FF0 + Fwhere Fmax, F0 and v0 are empirically determinedparameters. It turns out to be convenient to ex-press Hill’s equation in terms of the stress, σ =df FA ,and the strain rate ψ =df 1δt δxL ≡ vL .We can express the stress in terms of the strain rate,σ = σmaxψ0 − σ0ψψ0 + ψand use the result to calculate, for a given strainrate, the power output per unit volume of muscletissue:P = σ ψ .Different types of muscle tissue contract at differ-ent maximum rates6. The so-called “fast” musclesare (relatively) anærobic in their metabolism.(That is, they use a chemical reaction that doesnot require free oxygen to generate their energy.)Anærobic muscles contain fewer mitochondria(the sites of oxidative phosphorylation) and lowerdensity of myoglobin (myoglobin is related tohæmoglobin; it is used by muscle cells to storePhysics of the Human Body 25Chapter 3: Musculoskeletal system4. C.J. Pennycuick, Newton Rules Biology (Oxford U. Press, Oxford, 1992).5. A.V. Hill, Proc. Roy. Soc. Ser. B 126 (1938)