what amound of force to hold her leg in this position

Muscles, basic, and joints are some of the most interesting applications of statics. In that location are some surprises. Muscles, for example, exert far greater forces than nosotros might think. Figure 1 shows a forearm holding a book and a schematic diagram of an analogous lever arrangement. The schematic is a expert approximation for the forearm, which looks more complicated than it is, and we tin can go some insight into the way typical muscle systems function past analyzing it.

Muscles can simply contract, so they occur in pairs. In the arm, the biceps muscle is a flexor—that is, information technology closes the limb. The triceps muscle is an extensor that opens the limb. This configuration is typical of skeletal muscles, bones, and joints in humans and other vertebrates. Nigh skeletal muscles exert much larger forces within the trunk than the limbs apply to the outside world. The reason is clear once we realize that nigh muscles are attached to bones via tendons close to joints, causing these systems to have mechanical advantages much less than one. Viewing them as unproblematic machines, the input forcefulness is much greater than the output strength, equally seen in Figure 1.

Figure 1. (a) The figure shows the forearm of a person holding a volume. The biceps exert a force F_B to support the weight of the forearm and the volume. The triceps are assumed to exist relaxed. (b) Here, y'all can view an approximately equivalent mechanical system with the pivot at the elbow joint as seen in Example ane.

Example 1. Muscles Exert Bigger Forces Than Y'all Might Think

Calculate the force the biceps muscle must exert to hold the forearm and its load equally shown in Figure 1, and compare this force with the weight of the forearm plus its load. You may have the data in the figure to be accurate to 3 significant figures.

Strategy

At that place are 4 forces interim on the forearm and its load (the arrangement of interest). The magnitude of the force of the biceps is F _B; that of the elbow joint is F _E; that of the weights of the forearm is w _a, and its load is west _b. Two of these are unknown (F _B and F _E), then that the showtime condition for equilibrium cannot by itself yield F _B. Only if we utilise the second condition and choose the pin to be at the elbow, so the torque due to F _Eastward is zero, and the simply unknown becomes F _B.

Solution

The torques created by the weights are clockwise relative to the pivot, while the torque created by the biceps is counterclockwise; thus, the 2nd condition for equilibrium (cyberspace τ = 0) becomes

r _ii westward _a+r ₃ w _b=r ₁ F _B

Note that sin θ = ane for all forces, since θ= 90º for all forces. This equation can easily be solved for F _B in terms of known quantities, yielding

[latex]{F}_{\text{B}}=\frac{{r}_{2}{w}_{\text{a}}+{r}_{3}{west}_{\text{b}}}{{r}_{1}}\\[/latex].

Entering the known values gives

[latex]{F}_{\text{B}}=\frac{\left(0\text{.}\text{160}\text{m}\correct)\left(2\text{.}\text{50}\text{kg}\right)\left(9\text{.}\text{80}{\text{m/south}}^{2}\right)+\left(0\text{.}\text{380}\text{grand}\right)\left(four\text{.}\text{00}\text{kg}\correct)\left(9\text{.}\text{80}{\text{thousand/southward}}^{ii}\right)}{0\text{.}\text{0400}\text{ m}}\\[/latex]

which yields

F _B= 470 North.

Now, the combined weight of the arm and its load is (6.50 kg)(9.80 chiliad/s^two) = 63.7 N, and then that the ratio of the force exerted by the biceps to the total weight is

[latex]\frac{{F}_{\text{B}}}{{w}_{\text{a}}+{due west}_{\text{b}}}=\frac{\text{470}}{\text{63}\text{.}7}=7\text{.}\text{38}\\[/latex].

Discussion

This means that the biceps musculus is exerting a force 7.38 times the weight supported.

In the higher up example of the biceps muscle, the angle between the forearm and upper arm is ninety°. If this angle changes, the force exerted past the biceps muscle also changes. In addition, the length of the biceps muscle changes. The forcefulness the biceps muscle can exert depends upon its length; it is smaller when it is shorter than when information technology is stretched.

Very large forces are likewise created in the joints. In the previous example, the downward force F _E exerted by the humerus at the elbow joint equals 407 N, or 6.38 times the total weight supported. (The calculation of F _Eastward is straightforward and is left as an cease-of-chapter problem.) Considering muscles tin can contract, but not expand across their resting length, joints and muscles often exert forces that act in opposite directions and thus subtract. (In the above example, the upward force of the muscle minus the downwardly force of the joint equals the weight supported—that is, 470 North–407 Due north = 63 N, approximately equal to the weight supported.) Forces in muscles and joints are largest when their load is a long distance from the joint, as the volume is in the previous example.

In racquet sports such as lawn tennis the constant extension of the arm during game play creates large forces in this mode. The mass times the lever arm of a tennis racquet is an important gene, and many players apply the heaviest racquet they tin handle. It is no wonder that joint deterioration and damage to the tendons in the elbow, such every bit "tennis elbow," can upshot from repetitive motility, undue torques, and possibly poor racquet selection in such sports. Various tried techniques for holding and using a racquet or bat or stick non only increases sporting prowess but tin can minimize fatigue and long-term damage to the torso. For case, tennis balls correctly striking at the "sweetness spot" on the racquet will result in fiddling vibration or impact force being felt in the racquet and the trunk—less torque every bit explained in Collisions of Extended Bodies in 2 Dimensions. Twisting the hand to provide top spin on the ball or using an extended rigid elbow in a backhand stroke can also beal the tendons in the elbow.

Preparation coaches and concrete therapists use the knowledge of relationships between forces and torques in the treatment of muscles and joints. In physical therapy, an exercise routine can apply a detail strength and torque which can, over a menses of time, revive muscles and joints. Some exercises are designed to be carried out under water, considering this requires greater forces to be exerted, further strengthening muscles. However, connecting tissues in the limbs, such as tendons and cartilage too as joints are sometimes damaged past the large forces they carry. Often, this is due to accidents, only heavily muscled athletes, such as weightlifters, can tear muscles and connecting tissue through effort alone.

The back is considerably more complicated than the arm or leg, with diverse muscles and joints between vertebrae, all having mechanical advantages less than one. Back muscles must, therefore, exert very big forces, which are borne by the spinal column. Discs crushed by mere exertion are very common. The jaw is somewhat exceptional—the masseter muscles that shut the jaw have a mechanical advantage greater than 1 for the back teeth, allowing usa to exert very large forces with them. A crusade of stress headaches is persistent clenching of teeth where the sustained big force translates into fatigue in muscles around the skull.

Figure 2 shows how bad posture causes back strain. In function (a), we see a person with good posture. Notation that her upper body's cg is directly above the pivot signal in the hips, which in plough is directly higher up the base of operations of support at her feet. Because of this, her upper torso'southward weight exerts no torque about the hips. The only strength needed is a vertical force at the hips equal to the weight supported. No muscle action is required, since the bones are rigid and transmit this strength from the floor. This is a position of unstable equilibrium, but only small forces are needed to bring the upper body back to vertical if it is slightly displaced. Bad posture is shown in office (b); nosotros see that the upper body's cg is in front of the pin in the hips. This creates a clockwise torque around the hips that is counteracted by muscles in the lower back. These muscles must exert large forces, since they have typically pocket-size mechanical advantages. (In other words, the perpendicular lever arm for the muscles is much smaller than for the cg.) Poor posture can as well crusade muscle strain for people sitting at their desks using computers. Special chairs are available that let the body's CG to be more than easily situated to a higher place the seat, to reduce back hurting. Prolonged muscle action produces muscle strain. Annotation that the cg of the entire body is still direct above the base of support in part (b) of Effigy ii. This is compulsory; otherwise the person would non be in equilibrium. Nosotros lean forwards for the same reason when conveying a load on our backs, to the side when carrying a load in ane arm, and backward when conveying a load in forepart of u.s., as seen in Figure 3.

Figure ii. (a) good posture places the upper body'due south cg over the pivots in the hips, eliminating the demand for muscle activeness to residuum the body. (b) Poor posture requires exertion past the back muscles to counteract the clockwise torque produced around the pivot by the upper body'due south weight. The back muscles take a small effective perpendicular lever arm, r _b⊥, and must therefore exert a large force F _b. Notation that the legs lean backward to continue the cg of the entire body above the base of support in the feet.

You take probably been warned against lifting objects with your dorsum. This action, even more than bad posture, can cause muscle strain and impairment discs and vertebrae, since abnormally large forces are created in the back muscles and spine.

Figure 3. People adjust their stance to maintain balance. (a) A father carrying his son piggyback leans forrad to position their overall cg in a higher place the base of support at his feet. (b) A student conveying a shoulder bag leans to the side to go along the overall cg over his feet. (c) Another student carrying a load of books in her arms leans backward for the same reason.

Example ii. Practice Non Lift with Your Back

Consider the person lifting a heavy box with his back, shown in Effigy 4. (a) Summate the magnitude of the force F _B– in the back muscles that is needed to support the upper body plus the box and compare this with his weight. The mass of the upper body is 55.0 kg and the mass of the box is 30.0 kg. (b) Calculate the magnitude and direction of the force F ₅– exerted by the vertebrae on the spine at the indicated pin point. Again, information in the effigy may exist taken to be authentic to three pregnant figures.

Strategy

By now, nosotros sense that the 2d status for equilibrium is a good place to outset, and inspection of the known values confirms that it tin be used to solve for F _B– if the pin is chosen to be at the hips. The torques created by westward _ub and w _box– are clockwise, while that created past F _B– is counterclockwise.

Solution for (a)

Using the perpendicular lever artillery given in the figure, the 2d condition for equilibrium (netτ = 0) becomes

(0.350 m)(55.0 kg)(nine.lxxx m/s^two) + (0.500 m)(30.0 kg)(nine.eighty m/s^two) = (0.0800 thousand)F _B.

Solving for F _B yields

F _B= 4.xx × 10^three N.

The ratio of the forcefulness the dorsum muscles exert to the weight of the upper body plus its load is

[latex]\frac{{F}_{\text{B}}}{{westward}_{\text{ub}}+{w}_{\text{box}}}=\frac{\text{4200 Northward}}{\text{833 Northward}}=\text{5.04}\\[/latex].

This force is considerably larger than information technology would be if the load were non nowadays.

Solution for (b)

More important in terms of its damage potential is the force on the vertebrae F _V. The beginning condition for equilibrium (internet F = 0) can be used to find its magnitude and direction. Using y for vertical and x for horizontal, the condition for the net external forces along those axes to exist zilch

cyberspace F_y  = 0 and F_x = 0

Starting with the vertical ( y ) components, this yields

[latex]{F}_{\text{Five}y}-{due west}_{\text{ub}}-{w}_{\text{box}}-{F}_{\text{B}}\sin{29.0}^{\circ}=0\\[/latex]

Thus,

[latex]\begin{array}{lll}{F}_{\text{5}y}& =& {west}_{\text{ub}}+{w}_{\text{box}}+{F}_{\text{B}}\sin{29.0}^{\circ}\ & =& \text{833 North}+\left(\text{4200 Northward}\correct)\sin{29.0}^{\circ}\end{array}\\[/latex]

yielding

F _Vy = ii.87 × 10³ N.

Similarly, for the horizontal ( x ) components,

[latex]{F}_{\text{V}x}-{F}_{\text{B}}\cos{29.0}^{\circ}=0\\[/latex]

yielding

F _Vx = iii.67 × x³ N.

The magnitude of F_V is given by the Pythagorean theorem:

[latex]{{F}_{\text{V}}}=\sqrt{{{F}_{\text{V}x}}^{2}+{{F}_{\text{Five}y}}^{2}}=\text{four.66}\times {ten}^{3}\text{ N}\\[/latex].

The direction of F_V is

[latex]\theta =\tan^{-1}\left(\frac{{F}_{\text{V}y}}{{F}_{\text{V}x}}\right)=38.0^{\circ}\\[/latex].

Annotation that the ratio of F _V to the weight supported is

[latex]\frac{{F}_{\text{V}}}{{w}_{\text{ub}}+{w}_{\text{box}}}=\frac{\text{4660 North}}{\text{833 N}}=5.59\\[/latex].

Discussion

This strength is about v.6 times greater than it would be if the person were continuing erect. The trouble with the back is not and then much that the forces are large—because similar forces are created in our hips, knees, and ankles—but that our spines are relatively weak. Proper lifting, performed with the back erect and using the legs to enhance the body and load, creates much smaller forces in the back—in this case, well-nigh 5.6 times smaller.

Figure 4. This figure shows that large forces are exerted by the back muscles and experienced in the vertebrae when a person lifts with their dorsum, since these muscles have pocket-size effective perpendicular lever arms. The information shown here are analyzed in the preceding example, Instance 2.

What are the benefits of having nigh skeletal muscles attached so shut to joints? One reward is speed because small musculus contractions can produce big movements of limbs in a brusk menstruum of fourth dimension. Other advantages are flexibility and agility, made possible by the big numbers of joints and the ranges over which they function. For example, information technology is hard to imagine a organisation with biceps muscles attached at the wrist that would be capable of the broad range of movement we vertebrates possess.

There are some interesting complexities in existent systems of muscles, basic, and joints. For instance, the pivot point in many joints changes location as the articulation is flexed, so that the perpendicular lever arms and the mechanical advantage of the system change, likewise. Thus the forcefulness the biceps muscle must exert to hold up a volume varies as the forearm is flexed. Similar mechanisms operate in the legs, which explain, for example, why there is less leg strain when a bicycle seat is ready at the proper height. The methods employed in this section requite a reasonable clarification of existent systems provided enough is known nearly the dimensions of the system. In that location are many other interesting examples of force and torque in the body—a few of these are the subject of finish-of-chapter problems.

Section Summary

• Statics plays an important part in agreement everyday strains in our muscles and basic.
• Many lever systems in the body have a mechanical advantage of significantly less than one, as many of our muscles are attached shut to joints.
• Someone with good posture stands or sits in such as way that their eye of gravity lies directly above the pivot point in their hips, thereby avoiding back strain and harm to disks.

Conceptual Questions

1. Why are the forces exerted on the outside earth by the limbs of our bodies usually much smaller than the forces exerted by muscles inside the trunk?

2. Explain why the forces in our joints are several times larger than the forces we exert on the outside world with our limbs. Can these forces be even greater than muscle forces?

three. Certain types of dinosaurs were bipedal (walked on two legs). What is a proficient reason that these creatures invariably had long tails if they had long necks?

four. Swimmers and athletes during contest need to go through certain postures at the beginning of the race. Consider the residuum of the person and why start-offs are so important for races.

v. If the maximum force the biceps muscle can exert is chiliad Due north, can we option upward an object that weighs yard North? Explicate your answer.

6. Suppose the biceps muscle was attached through tendons to the upper arm shut to the elbow and the forearm about the wrist. What would be the advantages and disadvantages of this type of construction for the motion of the arm?

vii. Explain i of the reasons why pregnant women often suffer from back strain late in their pregnancy.

Exercises

one. Verify that the force in the elbow articulation in Case 1: Muscles Exert Bigger Forces Than You Might Call up (above) is 407 N, equally stated in the text.

2. Two muscles in the back of the leg pull on the Achilles tendon as shown in Figure 5. What total force exercise they exert?

Effigy five. The Achilles tendon of the posterior leg serves to attach plantaris, gastrocnemius, and soleus muscles to calcaneus os.

iii. The upper leg muscle (quadriceps) exerts a strength of 1250 Due north, which is carried by a tendon over the kneecap (the patella) at the angles shown in Figure 6. Notice the management and magnitude of the forcefulness exerted by the kneecap on the upper leg os (the femur).

Effigy 6. The articulatio genus joint works like a hinge to bend and straighten the lower leg. Information technology permits a person to sit, stand, and pivot.

4. A device for exercising the upper leg muscle is shown in Figure 7, together with a schematic representation of an equivalent lever system. Calculate the forcefulness exerted by the upper leg musculus to lift the mass at a constant speed. Explicitly show how you follow the steps in the Trouble-Solving Strategy for static equilibrium in Applications of Statistics, Including Problem-Solving Strategies.

Effigy 7. A mass is connected by pulleys and wires to the ankle in this exercise device.

5. A person working at a drafting board may agree her head as shown in Effigy 8, requiring muscle action to support the caput. The three major interim forces are shown. Calculate the direction and magnitude of the force supplied by the upper vertebrae F _V to hold the head stationary, assuming that this forcefulness acts along a line through the center of mass as do the weight and musculus forcefulness.

Figure viii.

6. We analyzed the biceps muscle example with the bending betwixt forearm and upper arm set at 90º. Using the same numbers equally in Example 1: Muscles Exert Bigger Forces Than You lot Might Recollect(above), discover the force exerted by the biceps muscle when the angle is 120º and the forearm is in a downward position.

7. Even when the head is held cock, as in Figure 9, its center of mass is not straight over the principal point of back up (the atlanto-occipital joint). The muscles at the back of the neck should therefore exert a forcefulness to keep the head erect. That is why your head falls forward when you lot fall asleep in the course. (a) Summate the force exerted past these muscles using the information in the figure. (b) What is the forcefulness exerted by the pivot on the caput?

Figure 9. The eye of mass of the head lies in forepart of its major point of support, requiring musculus action to agree the caput cock. A simplified lever organisation is shown.

8. A 75-kg human being stands on his toes past exerting an upward force through the Achilles tendon, as in Figure 10. (a) What is the force in the Achilles tendon if he stands on one foot? (b) Calculate the force at the pivot of the simplified lever system shown—that force is representative of forces in the talocrural joint joint.

Effigy 10.

9. A father lifts his child as shown in Figure 11. What force should the upper leg muscle exert to elevator the kid at a constant speed?

Effigy eleven. A child being lifted by a father'south lower leg.

10. Unlike most of the other muscles in our bodies, the masseter musculus in the jaw, as illustrated in Figure 12, is attached relatively far from the joint, enabling large forces to exist exerted by the back teeth. (a) Using the information in the figure, calculate the strength exerted by the lower teeth on the bullet. (b) Calculate the forcefulness on the joint.

Effigy 12. A person clenching a bullet between his teeth.

eleven. Integrated ConceptsSuppose we supercede the 4.0-kg book in Exercise 7 of the biceps muscle with an elastic exercise rope that obeys Hooke's Law. Assume its strength constant m = 600 N/m (a) How much is the rope stretched (past equilibrium) to provide the aforementioned force F _B as in this case? Presume the rope is held in the hand at the aforementioned location equally the book. (b) What force is on the biceps muscle if the exercise rope is pulled straight upwards so that the forearm makes an angle of 25º with the horizontal? Assume the biceps muscle is still perpendicular to the forearm.

12. (a) What force should the woman in Figure xiii exert on the floor with each hand to do a push-upwardly? Assume that she moves upwardly at a constant speed. (b) The triceps muscle at the back of her upper arm has an effective lever arm of i.75 cm, and she exerts force on the floor at a horizontal distance of xx.0 cm from the elbow articulation. Calculate the magnitude of the force in each triceps muscle, and compare it to her weight. (c) How much work does she do if her center of mass rises 0.240 m? (d) What is her useful power output if she does 25 pushups in one minute?

Effigy thirteen.

13. You have just planted a sturdy 2-1000-tall palm tree in your front lawn for your mother'south birthday. Your brother kicks a 500 1000 ball, which hits the pinnacle of the tree at a speed of five m/southward and stays in contact with it for ten ms. The ball falls to the ground near the base of the tree and the recoil of the tree is minimal. (a) What is the force on the tree? (b) The length of the sturdy section of the root is only 20 cm. Furthermore, the soil around the roots is loose and we tin can assume that an effective force is practical at the tip of the 20 cm length. What is the effective force exerted by the end of the tip of the root to keep the tree from toppling? Assume the tree volition be uprooted rather than bend. (c) What could yous accept done to ensure that the tree does not uproot easily?

14. Unreasonable ResultsSuppose 2 children are using a uniform seesaw that is 3.00 m long and has its center of mass over the pivot. The get-go child has a mass of 30.0 kg and sits one.xl yard from the pivot. (a) Calculate where the second eighteen.0 kg child must sit to balance the seesaw. (b) What is unreasonable near the outcome? (c) Which premise is unreasonable, or which premises are inconsistent?

xv. Construct Your Own ProblemConsider a method for measuring the mass of a person's arm in anatomical studies. The subject area lies on her back, extends her relaxed arm to the side and two scales are placed below the arm. One is placed under the elbow and the other under the dorsum of her hand. Construct a problem in which you summate the mass of the arm and discover its center of mass based on the calibration readings and the distances of the scales from the shoulder joint. You lot must include a free body diagram of the arm to direct the analysis. Consider changing the position of the scale under the manus to provide more information, if needed. You may wish to consult references to obtain reasonable mass values.

Selected Solutions to Bug & Exercises

1. [latex]\brainstorm{array}{lll}{F}_{\text{B}}&=&\text{470 Northward; }{r}_{1}=\text{4.00 cm; }{west}_{\text{a}}=\text{two.50 kg; }{r}_{2}=\text{16.0 cm; }{west}_{\text{b}}=\text{4.00 kg; }{r}_{3}=\text{38.0 cm}\\ {F}_{\text{E}}& =& {w}_{\text{a}}\left(\frac{{r}_{2}}{{r}_{i}}-i\right)+{w}_{\text{b}}\left(\frac{{r}_{3}}{{r}_{1}}-1\right)\\ & =& \left(\text{ii.50 kg}\correct)\left(9.fourscore\text{one thousand}/{\text{s}}^{2}\right)\left(\frac{\text{16.0 cm}}{\text{4.0 cm}}-1\correct)\\ & & +\left(\text{4.00 kg}\correct)\left(nine.80\text{m}/{\text{south}}^{2}\right)\left(\frac{\text{38.0 cm}}{\text{4.00 cm}}-1\right)\\ & =& \text{407 N}\terminate{array}\\[/latex]

3. 1.1 x x³ Due north, θ = 190 º ccw from positive x– centrality

5. F _V = 97 North, θ = 59º

seven. (a) 25 N downward (b) 75 Northward upwards

9. (a) F _A = ii.21 × 10^three N upward (b) F _B = 2.94 × 10^three North  downward

11. (a)F _{teeth on bullet} = ane.2 × 10² N upwardly (b) F _J = 84 N downward

xiii. (a) 147 N downwardly (b) 1680 North, 3.4 times her weight (c) 118 J (d) 49.0 West

15.  (a) [latex]{\bar{x}}_{2}=\text{2.33 m}\\[/latex] (b) The seesaw is iii.0 m long, and hence, there is only 1.fifty m of board on the other side of the pivot. The 2d child is off the board. (c) The position of the first child must be shortened, i.e. brought closer to the pin.

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