Muscles

Abstract

Overviews of the types of muscles in the body and their macroscopic structure are first presented, followed by models of muscle performance from macroscopic, microscopic, and nanoscopic perspectives. This includes analysis of forces generated by muscles as a function of their length and contraction speed, and how they depend on passive muscle materials and tendons. The discussion of the sliding filament model and sarcomeres includes the role of the elastic material titin. Muscle coordination, training, development, and fatigue are also addressed.

Problems

Muscles and Forces

5.1

There is an average normal vertical force of 1,710 N ($2.7 \times$ body weight) on the one foot of an athlete in contact with the ground during running, but an average of only 715 N on each foot (and 1,430 N total vertical force with both feet $= 2.3 \times$ body weight) when this same athlete does a vertical jump. Is this difference possible? Qualitatively why is it reasonable or unreasonable? (Consider only how the ranges of muscle lengths in these activities contribute to these observations.References [3] and [4] suggest that the stimulation of the muscles during stretching that occurs immediately prior to contraction in running, but not in jumping, contributes to this observation. This is also consistent with the observation that down-and-up action prior to a high jump leads to higher jumps.)

5.2

(a) Do the data presented in Fig. 5.46 suggest that the strength of muscles varies linearly with their cross-sectional areas, as is assumed in the text?

(b) If so, what is the proportionality constant?

(c) How does it differ for males and females?

Fig. 5.46

The strength of the arm flexor muscles for males and females versus muscle cross-sectional area. (Using data from [39].) For Problem 5.2

5.3

The fibers in the pectoralis major muscle converge on the proximal part of the humerus. With the humerus pointing down, the many fibers can be modeled as a clavicular component exerting a force of 224 N at an angle 0.55 rad above the horizontal and a sternal component (in the same plane and with the same timing) exerting a force of 251 N at an angle 0.35 rad below the horizontal. Find the resultant force and angle [16].

5.4

(a) Show that the physiological cross-sectional area (PCA) of a muscle for a parallel-fibered muscle is PCA $= (m/Ld)$ cm$^{2}$, where m is the mass of the muscle fibers, d is the density of muscles (1.056 g/cm$^{3}$), and L is the length of the muscle fibers.

(b) Repeat this for pinnate muscles, showing that PCA $= (m\cos \theta /Ld)$ cm$^{2}$, where $\theta$ is the pinnation angle (the angle between the long axis of the muscle and the fiber angle).

5.5

Check to see if the two relations in Problem 5.4 are consistent with the muscle data given in Table 5.4.

5.6

Let us estimate how much weight a person can lift above his head, with extended arms.

(a) Use the data for the 70 kg person in Chap. 1, and specifically the volume of skeletal muscle. Let us assume that all of these muscles are arranged as an upright right circular cylinder of height equal to the sum of the lengths of the person’s legs, torso, and arms. Use the resulting cross-sectional area and assume that skeletal muscle can exert a force of 20 N/cm$^{2}$ to find the mass that the person can lift.

(b) Is this value reasonable? If not, describe how each of the following factors may or may not contribute to an unreasonable result: (i) His arms are narrower than his legs, so the cross-sectional area of muscle is not uniform and the limiting factor would be the weakest (i.e., narrowest) link. (ii) The muscles are not the length assumed and should be taken to be no longer than a fraction of the length of an arm or leg segment. Consequently, there are several muscles in series. (iii) Not all skeletal muscle in the body is designed to contribute to this lifting. (iv) Not all of the muscles involved are parallel fibered muscles. (v) The 20 N/cm$^{2}$ value is inaccurate.

(c) Now re-evaluate the problem for female and male world-class weightlifters for their world-record lifted masses. (Use the data given in the text.) What fraction of their skeletal muscle cross-sectional area (assumed to be uniform here) effectively contributes to the lift?

5.7

Here are some data for world weightlifting records (in 2014 for clean-and-jerk) for people of different masses: for men, 169 kg lifted (by a 56 kg man), 182 kg (62 kg), 198 kg (69 kg), 210 kg (77 kg), 218 kg (85 kg), 233 kg (94 kg), and 238 kg (105 kg); and for women, 121 kg (by a 48 kg woman), 131 kg (53 kg), 141 kg (58 kg), 143 kg (63 kg), 158 kg (69 kg), and 163 (75 kg). How does the lifted weight vary with body weight? (Do this separately for men and women.) Does it follow a power law? Is it sublinear? How can you explain this dependence?

5.8

How can a measurement of force versus muscle length be made with the muscle fixed at either end (isometric conditions), as in Fig. 5.19? Is there an inconsistency here?

5.9

Let us say that a fusiform muscle of length L and PCA is attached to bones by tendons of length 0.9L and cross-section area f(PCA) ($f\ll 1$) on either end. This total length of 2.8L is kept fixed as the muscle contracts by 20% and develops a force corresponding to 15 N/cm$^{2}$. Use the data in Table 4.2.

(a) Find the strain in the tendons. How does it compare to the UPE for tendons?

(b) Relate f to Y for the tendon.

(c) Using the value of Y from the table, compare the diameters of the muscle and tendons.

(d) Find the stress in the tendons. How does it compare to the UPE for tendons?

5.10

(advanced problem) Show that the two models in Fig. 5.14 are mathematically equivalent, with $c_{1}/(k_{1}+k_{2})=c_{2}/k_{4}$; $k_{1}k_{2}/(k_{1}+k_{2})=k_{3}$; $k_{2}=k_{3}+k_{4}$; and $(k_{2}/(k_{1}+k_{2}))T_\mathrm{G1}=T_\mathrm{G2}$ [53].

5.11

The maximum power that can be obtained from skeletal muscles is estimated to be $\sim$250–500 W/kg for isotonic conditions and $1.75\times$ this with pre-stretching [25]. Assume that during baseball pitching the work done during humerus rotation is $\sim$132 J over 0.034 s. The shoulder muscles involved in this rotation are thought to have a mass as high as of 2.16 kg in large people.

(a) What is the mechanical power associated with this shoulder rotation?

(b) Determine this power per unit mass of shoulder muscle and compare it to the maximum power derivable from pre-stretched muscle.

(c) If the power per mass available from the muscle turns out to be smaller than this measured increase in kinetic energy per unit time, explain how this could be so [63].

5.12

Repeat Problem 5.11, now including the spreads in performance as given by the standard deviations, with work done during humerus rotation being $132 \pm 52$ J over $0.034 \pm 0.013$ s [63].

5.13

It is known that maximum muscle forces are $1.75\times$ larger when the muscles are pre-stretched than under isotonic conditions. In the text and in Problem 5.11, it was assumed that this same factor applies when comparing the maximum specific power from muscles for these two conditions. What does this assume?

Muscle Tension Versus Length or Time

5.14

Why do you suppose that Fig. 3.54 plots the total length of the muscles and tendons?

5.15

Use the data in Fig. 3.54 to determine whether the major leg muscles used during bicycling are always near their optimal lengths (so they always exert nearly maximum forces) or do they become much longer or shorter than these optimal lengths during parts of the cycle (so they exert forces that are much less than the maxima)? Justify your answer.

5.16

In the example of throwing a ball studied in Chap. 3, we assumed that the force developed by the biceps brachii did not depend on the angle between the upper and lower arm; this angle varied from 180–$0^{\circ }$ in the first statement of the problem and from 135–$45^{\circ }$ in the second. Use what we now know about how the contractile force of a muscle varies with its length to determine the validity of that second assumption. Assume that the muscle has a length of $L_{0} = 12$ cm when resting at $90^{\circ }$, and that the distance from the elbow joint to the insertion point on the radius is 4 cm (see Fig. 5.47). (You should base your conclusions on calculations of the muscle length for various angles—at least for 0, 45, 90, 135, and $180^{\circ }$. Ignore the length of the tendons.)

5.17

In fusiform (or parallel) muscles all of the muscle contractile force is directed along the axis of the tendon and there is a relationship between changes in muscle and tendon length that depends on conditions (isometric contraction, etc.). In pinnate and bipinnate muscles these relationships are qualitatively different. Refer to Fig. 5.48, which depicts a bipinnate muscle in relatively more relaxed, initial (solid lines) and relatively more contracted, final (dashed lines) configurations.

(a) In the initial configuration, the tendon length (to the point where the muscle is inserted into the tendon, at point B) is 8.7 cm ($=2.9$ cm + 5.8 cm in the figure) and each muscle fiber is 10 cm long, and is attached to the tendon at a $30^{\circ }$ angle. Say that there are a total of N fibers attached to the tendon near this point (with an equal number of fibers on both sides) and that each fiber exerts a force $F_{ \mathrm {fiber}}$. Find the total force transmitted to the tendon $F_{\mathrm { tendon}}$. (Is there “wasted” force here?)

(b) The muscle contracts so that the tendon length (to point C) is 2.9 cm, and the muscle/tendon angle is now $60^{\circ }$. Find the length of the muscle (see the dashed lines). During this contraction, which has become shorter more, the muscle fibers or the tendon? Is this good or bad for muscular action? (Does this effect, along with the ability to attach more muscle fibers to this type of tendon counter the “wasted” force in part (a)?)

(c) Find the force transmitted to the tendon after the contraction in part (b) $F_{\mathrm {tendon}}$, assuming that the contractile force of each muscle fiber is still $F_{\mathrm {fiber}}$. How has the force transmitted to the tendon changed?

(d) Now consider the more general case, in which we acknowledge that the force exerted by a muscle fiber changes with length. Say that $F_{\mathrm {fiber }}$ peaks at a length $L= 8$ cm, with $F_{\mathrm {fiber}}(L)=F_{\mathrm {max} }(1-(L-8)^{2}/8)$. Sketch $F_{\mathrm {tendon}}$ as L contracts from 10 to 6 cm. Does it peak exactly at the length where $F_{\mathrm {fiber}}(L)$ peaks? Why?

Fig. 5.47

Model of parallel fibered muscle. For Problem 5.16

Fig. 5.48

Model of bipinnate fibered muscle. (Based on [52].) For Problem 5.17

5.18

Let us model the decrease of muscle force F from its maximum value $F_{\mathrm {max}}$ at a length $L_{\mathrm {peak}}$ as being parabolic: $F(L)=F_{ \mathrm {max}}-\varDelta F((L-L_{\mathrm {peak}})/L_{\mathrm {dec}})^{2}$. Show that the work done by the muscle as it contracts from $L=L_{\mathrm {peak}}+L_{\mathrm { dec}}$ to $L=L_{\mathrm {peak}}-L_{\mathrm {dec}}$ is $W=W_{0}(1-\varDelta F/3F_{ \mathrm {max}})$, where $W_{0}=2F_{\mathrm {max}}L_{\mathrm {dec}}$ is the work that the muscle would do if the force did not change with length [55].

5.19

Sketch on one set of axes the six curves in Fig. 5.20, each as the $\%$ of maximum strength for that muscle group versus the angle of pull (which is the joint angle). The ordinate axis should range from 0 to 100% (and of course $100\%$ represents a different maximum force for each of the curves).

5.20

(a) Show that the model we used for the torque that can develop in a joint due to muscle contraction, $\tau =\tau _{\mathrm {max}}((\omega _{\mathrm {max}} -\omega )/(\omega _{\mathrm {max}}+G))$, can be derived from the Hill force-velocity curve (5.21 ) . This is expressed as a function of the rate of change of joint angle during flexion, with $\omega = -d\theta /dt$, so $\omega$ is positive when the angle is becoming smaller, and where $\omega _{\mathrm {max}}$ and $G\sim 3$ are constants. (We used this in the force models of the high and long jumps in Chap. 3.)

(b) Explain how the relation used for joint extension in this Chap. 3 model, $\tau =\tau _{\mathrm {max}}$, is a bit different than the one expected from the discussion in this chapter.

5.21

Determine the values of T and v that maximize the power output of a muscle described by the Hill force–velocity equation (5.20) and find the power output $P_{\mathrm {max}}$ of muscle. Take $a/T_{\mathrm {max}}=0.25$; remember that $P=Tv$. (Hint: You should differentiate the expression for power with respect to v and set it equal to zero. One way to do this is to first solve for T(v) from the Hill equation and express $P(v)=T(v)v$. The maximum power should turn out to be about $0.1 T_{\mathrm {max}}v_{\mathrm {max}}$.)

5.22

Use the Hill model for a muscle fiber (5.24) to show that T(t) evolves as $-T-( T_{\mathrm {max}}+a) \ln ((T_{\mathrm {max}}-T)/T_{\mathrm {max}}) =kbt$, for $\mathrm{d}L/\mathrm{d}t = 0$ (isometric condtions), when $T(0)=0$ and $T_{\mathrm {max}}$ is constant [41]. Show that T eventually becomes $T_{\mathrm {max}}$.

5.23

Use the Hill model for a muscle fiber (5.24) to show that T(t) evolves as $T_{\mathrm {max}}-T+(T_{u}+a) \ln ((T_{\mathrm {max}}-T_{u})/(T-T_{u})) =kt(b+u)$, where $(T_{u}+a)u=b(T_{\mathrm {max}}- T_{u})$, for a muscle fiber initially with $T(0)= T_{\mathrm {max}}$ that is then allowed to contract at constant speed $u=-\mathrm{d}L/\mathrm{d}t$ [41]. ($T_{\mathrm {max}}$ does not change.) Show that T eventually becomes $T_{u}$.

5.24

Use the Hill model for a muscle fiber (5.24) to determine the tension in a muscle fiber, initially at $T_{\mathrm {max}}$ that is suddenly shortened from its initial value of $L{_{0}}$ by $L{_{1}}$ at $t = 0$ [41].

(a) Show that $L(t) = L{_{0}} - L{_{1}}(1- \theta (t))$ and so $\mathrm{d}L/\mathrm{d}t = -L{_{1}} \delta (t)$ and (5.24) becomes $\mathrm{d}T/\mathrm{d}t = k(-L{_{1}} \delta (t)+b(T_{\mathrm {max}}-T)/(T+a))$. (This uses the Heaviside step and Dirac delta functions introduced in Chap. 4.)

(b) Integrate this from just before $t = 0$ to just after it (t from $0^{-}$ to $0^{+}$), to show that $T(0^{+})-T(0^{-}) = - k L{_{0}}$, which means that the consequence of the shortening is to decrease the initial tension from $T_{\mathrm {max}}$ to $T_{\mathrm {max}}- k L{_{1}}$, which we now call T(0).

(c) Show that for $t > 0$ (5.24) becomes $\mathrm{d}T/\mathrm{d}t = kb(T_{\mathrm {max}}-T)/(T+a)$, and integrate it with $T(0) = T_{\mathrm {max}}- k L{_{1}}$ to find T(t). Assume that $T_{\mathrm {max}}$ does not change with time. (This gives an approach to the final tension of $T_{\mathrm {max}}$ that is faster than observed by experiments, even with optimal curve fitting, and this illustrates one of the limitations of the Hill force-velocity curve.)

5.25

In the example of throwing a ball studied in Chap. 3, we assumed that the force developed by the biceps brachii did not depend on how fast the elbow angle $\theta$ varies with time. Figure 5.28 shows that this is not true, because the muscle length and contraction speed can both be important and can limit the tension developed in the muscle. One way to address this is by re-analyzing throwing, limiting muscle tension as indicated in this figure. Explain how this can be done numerically.

5.26

In the analysis of throwing a ball studied in Chap. 3 , we assumed that the force developed by the biceps brachii may depend on muscle length but not on how fast the elbow angle $\theta$ varied with time $(\mathrm{d}\theta /\mathrm{d}t)$. We analyze the limitations of that model by adopting an approach that is quite different from that used in Problem 5.25 (and we will not use Fig. 5.28 explicitly). Say $T_{\mathrm {max}} = (20$ N/cm$^{2}$)PCA, where PCA is the physiological cross-sectional area of a muscle, and that $v_{\mathrm {max}} = 6 L_{0}/$ s for muscles predominantly comprised of ST muscle fibers and $v_{\mathrm {max}} = 16 L_{0}/$ s for muscles predominantly comprised of FT muscle fibers, where $L_{0}$ is the optimal fiber length of the muscle of interest. (Because muscles are often composed of combinations of FT and ST muscle fiber, the appropriate $v_{\mathrm {max}}$ is often in between these limiting values.) Also assume that $a/T_{\mathrm {max}}=b/v_{\mathrm {max}}= 0.25$ and use the results from Problem 5.21. For each part that follows, consider the case studied in Chap. 3 of 2 in diameter biceps brachii muscles, for which the calculated throwing speed is 17.8 mph (with no gravity and $\langle \sin \theta \rangle \,=\,1$), and take $L_{0} = 10$ cm.

(a) Find a and b, first assuming FT and then ST muscles.

(b) If the ball leaves the hand at a speed of x mph, geometry says that the speed of muscle contraction is smaller by the proportion of the distance of muscle insertion from the elbow pivot (4 cm) to the distance of the ball from the pivot (36 cm). How does this speed of muscle contraction, based on the throwing calculation, compare to the maximum muscle contraction speed ($v_{\mathrm {max}}$) for FT and ST muscles? Also, how does it compare to the muscle contraction speed, for both types of muscles, at which the power generated by the muscle is maximized? (Ignore the response of the tendons. Can these be important?)

(c) The average and peak powers needed to be generated by the muscles to achieve these throwing speeds are 178 W and 356 W, respectively, for the 2 in diameter biceps using the kinematics calculations performed in Chap. 3. For both FT and ST muscles, calculate the maximum power that the muscle can generate and compare your answers with the values calculated using kinematics. Repeat this if you (incorrectly) assume that the force generated by the muscle is $T_{\mathrm {max} }$ independent of muscle contraction speed, so that the maximum muscle power would be $T_{\mathrm {max}}v_{\mathrm {max}}$ (which is clearly incorrect).

(d) Do the results in (b) and (c) cast doubt on the calculation in Chap. 3 (and why)? If so, does this totally invalidate the calculation or does it mean that after a certain muscle contraction speed is achieved the decrease in the muscle force must be included to improve the model? (Also, note that since the biceps brachii have much ST muscle fiber, the “FT” limit is not very realistic.)

Microscopic and Nanoscopic Processes in Muscles

5.27

(a) Estimate the time it takes Ca$^{2+}$ ions to diffuse throughout a typical skeletal muscle cell if the cell has a diameter of $200 \,\upmu$m and the diffusion coefficient of the ions is $10^{-5}$ cm$^{2}/$s. Assume that the release of ions occurs just outside of the cell, due to a nerve pulse arriving there, and diffusion occurs in one transverse dimension of the cell.

(b) How does this time compare to the typical reaction times of muscles?

(c) The nerve signal actually activates the sarcoplasmic reticulum that runs transverse to the outer membrane of the cell, in the Z-line borders of each sarcomere, and the Ca$^{2+}$ ions are released there; consequently, the maximum distance the ions need to travel to diffuse across the whole sarcomere is $2 \,\upmu$m. How long does this take and does this time seem more reasonable?

5.28

Estimate the number of sarcomeres in your biceps brachii:

(a) along its length,

(b) across its cross-section, and

(c) in total.

5.29

When lifting a weight, a muscle goes from being 25% shorter than its resting length to 25% longer than it. If the resting length of the sarcomere is $2.5 \,\upmu$m, how many 11-nm crossbridge power strokes occur in each sarcomere during the lift?

5.30

Figure 5.33 is the force versus displacement curve of a single myosin molecule interacting with an actin molecule during a powerstroke, as measured by “optical tweezers.” Since work is the integral of force over distance, estimate the work done by the myosin molecule during one powerstroke. (Use the dotted line and find the area in the triangle.) If the energy available from ATP hydrolysis in a muscle cell is $10^{-19}$ J, calculate the efficiency of a myosin powerstroke in using the energy from ATP hydrolysis. This estimate uses the mean force. The efficiency is ${\sim }1.75 \times$ larger if a less conservative estimate is made using the highest forces measured [17].

5.31

(a) If in a single actin–myosin crossbridge 4 pN is generated per crossbridge and the power stroke distance is 11 nm, as in Fig. 5.33, show that the mechanical work done is 22 pN–nm. Express this in J and eV (1 eV (electron volt) $= 1.6 \times 10^{-19}$ J) per crossbridge, and in kcal/mole (for a mole of crossbridges).

(b) If the hydrolysis of ATP releases 14 kcal/mole (as is seen in Chap. 6) and the hydrolysis of one ATP molecule activates one crossbridge, what fraction of the available energy is used in mechanical work?

5.32

(advanced problem) In the Huxley sliding filament model, assume $f(x) = 0$ for $x < 0$, $f _{\mathrm {1}} x/h$ for $0 \le x \le h$, and 0 for $x> h$, and $g(x) = g _{\mathrm {2}}$ for $x \le 0$ and $g _{\mathrm {1}} x/h$ for $x > 0$. Use (5.34) to show that $T_{\mathrm {max}} = Nkh^{2} F_{\mathrm {1}}/2$, where $F_{\mathrm {1}}=f _{\mathrm {1}}/(f _{\mathrm {1}} + g_{\mathrm {1}})$ [41].

5.33

(advanced problem) Use the Huxley sliding filament model (5.33) in steady state with constant contraction speed v, and f(x) and g(x) and the other notation from Problem 5.32, to show that for these rate functions $n(x) = F_{\mathrm {1}}(1- \exp (-V/v)) \exp [(2x/h) G_{\mathrm {2}}(V/v)]$ for $x < 0$, $= F_{\mathrm {1}}[1-\exp ([(x ^{2}/h ^{2}) - 1](V/v))]$ for $0 < x< h$, and $= 0$ for $x > h$, where the constant $V=( f _{\mathrm {1}} + g _{\mathrm {1}})h/2$, which has units of speed, and $G_{\mathrm {2}}=g _{\mathrm {2}}/( f _{\mathrm {1}} + g_{\mathrm {1}})$ [41]. (Hint: Do this by separately solving the Huxley equation in the three spatial regions, using the appropriate forms of the rate functions, to arrive at separate solutions and then choose the undetermined constants to make n(x) continuous at $x=0$ and $x=h$. Keep only solutions that do not go to infinity in the region (i.e. are bounded) within each range (and this means that the solution for $x > h$ is $n = 0$). For the middle region use a trial function $n(x) = A + B \exp (C x^{2})$.)

5.34

(advanced problem) Use n (x) and other notation from the solution to the Huxley sliding filament model for the chosen rate functions in Problem 5.33 to show that (5.32) becomes $T(v) = T_{\mathrm {max}}[1-(V/v)[1-\exp (-(V/v))](1+(v/V)/2 G_{\mathrm {2}}^{2})]$ [41]. This is the force-velocity equation that can reproduce the Hill force-velocity curve.

5.35

(advanced problem)

(a) In the Huxley sliding filament model prediction for T(v) from Problem 5.34, T ranges from $T_{\mathrm {max}}$ for $v=0$ to 0 for a given $v= v_{\mathrm {max}}$. Show that if $F_{\mathrm {1}}=0.812$ and $G_{\mathrm {2}}=3.919$, then $v_{\mathrm {max}}\simeq 4V$.

(b) Since $v_{\mathrm {max}}=b T_{\mathrm {max}}/a$ in the Hill force-velocity curve, show that with $T_{\mathrm {max}}= 4a$ (which is a common fit), then $V\simeq b$ in this application of the Huxley model [41]. This shows how the Huxley sliding filament model reproduces the limits of the Hill curve, and it is easy to show that it gives the same curve shape in between these two limits.

5.36

(advanced problem)

(a) For $f _{\mathrm {1}}= 65$/s, $g _{\mathrm {1}} = 15$/s, and $h = 10\,$nm, show that the Huxley model results in Problem 5.35 predict a maximum shortening speed of 1600 nm/s for each filament pair, which means for each half sarcomere, or 3200 nm/s for each sarcomere [41].

(b) If the sarcomeres are 2.5 $\upmu$m long and the muscle length is 1 cm, find how fast the muscle can contract.

5.37

(advanced problem) Plot the Huxley model results for T(v) from Problem 5.35 on the same axes as the Hill force-velocity curve and compare them.

Muscle Energy, Power, Synergy, Training, and Endurance

5.38

By using reasonable estimates of muscle length, speed and force per unit area, determine whether the estimate that $\sim$250 W are generated per kg of active muscle is reasonable.

5.39

A 75 kg person climbs stairs to a height of 50 m in 53 s. Determine the person’s metabolic rate during climbing, in W and kcal/h, assuming the process in $25\%$ efficient. How does this relate to the estimates of energy used in such motion in the text? (See Chap. 6 for more on metabolism.)

5.40

In a weight room, you exercise your biceps brachii with a constant weight by trying to lift it (i.e., rotating your lower arm about your elbow). Sketch the maximum tension versus length for your biceps brachii if

(a) The weight is so heavy that you can barely hold it as you try to lift it.

(b) The weight is so light that you can easily lift it.

For both cases, where is the weight on your sketch? Describe howis also an exampleis also an example much you are changing the length of your muscle during the “lift.”

5.41

During a forearm curl exercise, your upper arm is fixed, say at 45$^{\circ }$ to the vertical and your elbow is fixed, as you rotate your forearm (and hand) about your elbow (decreasing and later increasing the elbow angle), while holding a weight in your hand.

(a) Draw diagrams showing the directions and magnitudes of the forces due to the weight and your lower arm, both near the beginning of the first phase of the curl (elbow flexion, beginning with elbow angle 180$^{\circ }$) and the second phase of the curl (elbow extension, beginning with 25$^{\circ }$), first ignoring the muscles that flex the elbow [16].

(b) Labeling forces and distances as needed, find the torque your arm muscles need to supply for both cases, first assuming only equilibrium.

(c) Calculate these two torques for a 160 lb, 6 ft standard human and a dumbbell weight of 30 lb.

(d) Would you expect the torques you would need to apply during a curl (with actual motion) to be greater than, the same as, or less than these values?

5.42

(a) Draw a force diagram for the forearm curl exercise as in Problem 5.41, with your hand still holding a weight such as a barbell or dumbbell (constant-load training), and determine and sketch the torque about the elbow your muscles need to supply to maintain equilibrium as a function of elbow angle. In constant-load training the magnitude and direction of the force due to the free weight do not change during the exercise cycle.

(b) Repeat (a) for a variable load exercise machine in which a force is applied (at the hand) normal to the forearm that can vary with angle (variable-load training) [16, 54]. What should that variable load be as a function of angle to maintain equilibrium as a function of elbow angle (and at static equilbrium)? How does your answer differ from that in (a)?

(c) The maximum amount of torque your muscles can supply depends on their length and consequently on the joint angle. In training you often want to push the limits of your muscles to some extent. Are the features of variable-load training preferable to those of constant-load training for weight training? Why?

5.43

Fig. 5.37 shows that as a cat walks faster his gastrocnemius develop more and more force, but his soleus muscles do not. Explain why the relative contribution of the gastrocnemius muscle increases with speed.

5.44

You repetitively extend your arm and then flex it (say as to touch your shoulder), with a period of 2 s. Sketch the forces in your biceps brachii and triceps brachii each versus time, and both versus each other (with time as an implicit parameter).

5.45

Derive a hyperbolic expression that describes the Rohmert curve (Fig. 5.45) relating between the endurance time and exertion level of a muscle.