(Figure 1) Shows Two Forces Acting on an Object at Rest You May Want to Review

nine Statics and Torque

65 9.6 Forces and Torques in Muscles and Joints

Summary

• Explain the forces exerted by muscles.
• State how a bad posture causes back strain.
• Talk over the benefits of skeletal muscles fastened close to joints.
• Discuss diverse complexities in the real organization of muscles, bones, and joints.

Muscles, bones, and joints are some of the nigh interesting applications of statics. There are some surprises. Muscles, for instance, exert far greater forces than nosotros might retrieve. Figure 1 shows a forearm belongings a book and a schematic diagram of an analogous lever system. The schematic is a good approximation for the forearm, which looks more complicated than it is, and we tin can become some insight into the way typical muscle systems role by analyzing it.

Muscles can only contract, then they occur in pairs. In the arm, the biceps muscle is a flexor—that is, information technology closes the limb. The triceps musculus is an extensor that opens the limb. This configuration is typical of skeletal muscles, bones, and joints in humans and other vertebrates. About skeletal muscles exert much larger forces inside the body than the limbs utilize to the outside world. The reason is clear once we realize that nearly muscles are attached to bones via tendons close to joints, causing these systems to take mechanical advantages much less than ane. Viewing them as elementary machines, the input force is much greater than the output forcefulness, as seen in Figure ane.

Figure 1. (a) The effigy shows the forearm of a person belongings a book. The biceps exert a force F_B to back up the weight of the forearm and the book. The triceps are causeless to be relaxed. (b) Hither, yous can view an approximately equivalent mechanical arrangement with the pivot at the elbow joint as seen in Instance 1.

Example ane: Muscles Exert Bigger Forces Than You Might Recollect

Summate the forcefulness the biceps muscle must exert to concur the forearm and its load equally shown in Figure 1, and compare this force with the weight of the forearm plus its load. Y'all may take the data in the figure to be authentic to iii significant figures.

Strategy

In that location are 4 forces acting on the forearm and its load (the system of involvement). The magnitude of the force of the biceps is[latex]\boldsymbol{F_{\textbf{B}}};[/latex]that of the elbow articulation is[latex]\boldsymbol{F_{\textbf{E}}};[/latex]that of the weights of the forearm is[latex]\boldsymbol{w_{\textbf{a}}},[/latex]and its load is[latex]\boldsymbol{w_{\textbf{b}}}.[/latex]Two of these are unknown ([latex]\boldsymbol{F_{\textbf{B}}}[/latex]and[latex]\boldsymbol{F_{\textbf{E}}}[/latex]), then that the first status for equilibrium cannot by itself yield[latex]\boldsymbol{F_{\textbf{B}}}.[/latex]But if we use the 2d condition and choose the pivot to be at the elbow, then the torque due to[latex]\boldsymbol{F_{\textbf{E}}}[/latex]is null, and the only unknown becomes[latex]\boldsymbol{F_{\textbf{B}}}.[/latex]

Solution

The torques created by the weights are clockwise relative to the pivot, while the torque created by the biceps is counterclockwise; thus, the second condition for equilibrium[latex]\boldsymbol{(\textbf{ net }\tau=0)}[/latex]becomes

[latex]\boldsymbol{r_2w_{\textbf{a}}+r_3w_{\textbf{b}}=r_1F_{\textbf{B}}}.[/latex]

Note that[latex]\boldsymbol{\sin\theta=ane}[/latex]for all forces, since[latex]\boldsymbol{\theta=90^0}[/latex]for all forces. This equation tin can easily be solved for[latex]\boldsymbol{F_{\textbf{B}}}[/latex]in terms of known quantities, yielding

[latex]\boldsymbol{F_{\textbf{B}}\:=}[/latex][latex]\boldsymbol{\frac{r_2w_{\textbf{a}}+r_3w_{\textbf{b}}}{r_1}}.[/latex]

Entering the known values gives

[latex]\boldsymbol{F_{\textbf{B}}\:=}[/latex][latex]\boldsymbol{\frac{(0.160\textbf{ m})(2.50\textbf{ kg})(9.80\textbf{ m/s}^two)+(0.380\textbf{ m})(iv.00\textbf{ kg})(nine.80\textbf{ m/southward}^2)}{0.0400\textbf{ m}}}[/latex]

which yields

[latex]\boldsymbol{F_{\textbf{B}}=470\textbf{ N}}.[/latex]

At present, the combined weight of the arm and its load is[latex]\boldsymbol{(six.50\textbf{ kg})(ix.80\textbf{ k/s}^ii)=63.7\textbf{ N}},[/latex]so that the ratio of the force exerted by the biceps to the total weight is

[latex]\boldsymbol{\frac{F_{\textbf{B}}}{w_{\textbf{a}}+w_{\textbf{b}}}}[/latex][latex]\boldsymbol{=}[/latex][latex]\boldsymbol{\frac{470}{63.7}}[/latex][latex]\boldsymbol{=seven.38.}[/latex]

Word

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

In the higher up case of the biceps muscle, the angle betwixt the forearm and upper arm is 90°. If this angle changes, the force exerted by the biceps muscle besides changes. In add-on, the length of the biceps muscle changes. The force 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 also created in the joints. In the previous example, the downwards strength[latex]\boldsymbol{F_{\textbf{Eastward}}}[/latex]exerted by the humerus at the elbow articulation equals 407 N, or 6.38 times the total weight supported. (The calculation of[latex]\boldsymbol{F_{\textbf{Due east}}}[/latex]is straightforward and is left as an end-of-chapter problem.) Because muscles tin contract, but not expand across their resting length, joints and muscles often exert forces that deed in contrary directions and thus subtract. (In the above case, the upwards force of the muscle minus the downwardly force of the joint equals the weight supported—that is,[latex]\boldsymbol{470\textbf{ N}-407\textbf{ North}=63\textbf{ N}},[/latex]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 book is in the previous example.

In racquet sports such as tennis the abiding extension of the arm during game play creates large forces in this fashion. The mass times the lever arm of a lawn tennis racquet is an of import factor, and many players use the heaviest racquet they tin can handle. It is no wonder that articulation deterioration and harm to the tendons in the elbow, such as "lawn tennis elbow," can event from repetitive motion, undue torques, and possibly poor racquet selection in such sports. Various tried techniques for holding and using a racquet or bat or stick not but increases sporting prowess but can minimize fatigue and long-term harm to the torso. For example, tennis balls correctly hit at the "sweet spot" on the racquet volition effect in little vibration or impact force being felt in the racquet and the body—less torque as explained in Chapter 10.half dozen Collisions of Extended Bodies in Ii Dimensions. Twisting the paw to provide top spin on the ball or using an extended rigid elbow in a backhand stroke can also aggravate the tendons in the elbow.

Preparation coaches and physical therapists apply the knowledge of relationships betwixt forces and torques in the treatment of muscles and joints. In concrete therapy, an exercise routine can utilise a detail forcefulness and torque which can, over a menstruum of time, revive muscles and joints. Some exercises are designed to exist carried out under h2o, because this requires greater forces to be exerted, farther strengthening muscles. However, connecting tissues in the limbs, such as tendons and cartilage besides as joints are sometimes damaged by the big forces they carry. Frequently, this is due to accidents, merely heavily muscled athletes, such as weightlifters, tin can tear muscles and connecting tissue through effort alone.

The dorsum is considerably more complicated than the arm or leg, with various muscles and joints betwixt vertebrae, all having mechanical advantages less than 1. Back muscles must, therefore, exert very large 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, assuasive u.s. to exert very big forces with them. A cause of stress headaches is persistent clenching of teeth where the sustained large force translates into fatigue in muscles effectually the skull.

Effigy two shows how bad posture causes dorsum strain. In part (a), we encounter a person with proficient posture. Notation that her upper body's cg is directly in a higher place the pivot point in the hips, which in turn is direct above the base of support at her feet. Because of this, her upper body'due south weight exerts no torque about the hips. The but force needed is a vertical force at the hips equal to the weight supported. No muscle activeness is required, since the bones are rigid and transmit this force from the flooring. This is a position of unstable equilibrium, just merely small forces are needed to bring the upper body dorsum to vertical if information technology is slightly displaced. Bad posture is shown in part (b); nosotros see that the upper trunk'due south cg is in front of the pivot in the hips. This creates a clockwise torque around the hips that is counteracted by muscles in the lower back. These muscles must exert big forces, since they have typically small mechanical advantages. (In other words, the perpendicular lever arm for the muscles is much smaller than for the cg.) Poor posture can also cause muscle strain for people sitting at their desks using computers. Special chairs are available that allow the body's CG to be more easily situated above the seat, to reduce back pain. Prolonged muscle action produces muscle strain. Note that the cg of the entire body is withal direct to a higher place the base of back up in part (b) of Figure 2. This is compulsory; otherwise the person would not be in equilibrium. Nosotros lean forward for the same reason when carrying a load on our backs, to the side when carrying a load in one arm, and astern when carrying a load in front of us, equally seen in Figure iii.

Figure 2. (a) Good posture places the upper body's cg over the pivots in the hips, eliminating the need for muscle action to rest the body. (b) Poor posture requires exertion by the dorsum muscles to counteract the clockwise torque produced around the pivot past the upper body'due south weight. The back muscles have a minor effective perpendicular lever arm, r _b⊥ , and must therefore exert a large force F_b . Note that the legs lean backward to continue the cg of the unabridged body above the base of back up in the anxiety.

Y'all have probably been warned against lifting objects with your dorsum. This action, fifty-fifty more than bad posture, can cause musculus strain and impairment discs and vertebrae, since abnormally big forces are created in the dorsum muscles and spine.

Figure iii. People conform their stance to maintain rest. (a) A male parent carrying his son piggyback leans forward to position their overall cg above the base of back up at his feet. (b) A student conveying a shoulder bag leans to the side to keep the overall cg over his anxiety. (c) Another student carrying a load of books in her artillery leans backward for the same reason.

Example two: Do Non Lift with Your Back

Consider the person lifting a heavy box with his back, shown in Figure four. (a) Calculate the magnitude of the forcefulness[latex]\boldsymbol{F_{\textbf{B}}}[/latex]– in the back muscles that is needed to support the upper torso plus the box and compare this with his weight. The mass of the upper trunk is 55.0 kg and the mass of the box is 30.0 kg. (b) Calculate the magnitude and direction of the forcefulness[latex]\boldsymbol{F_{\textbf{V}}}[/latex]– exerted by the vertebrae on the spine at the indicated pin indicate. Again, data in the effigy may exist taken to be accurate to three significant figures.

Strategy

By now, we sense that the second condition for equilibrium is a good place to start, and inspection of the known values confirms that it tin exist used to solve for[latex]\boldsymbol{F_{\textbf{B}}}[/latex]– if the pin is called to be at the hips. The torques created by[latex]\boldsymbol{w_{\textbf{ub}}}[/latex]and[latex]\boldsymbol{w_{\textbf{box}}}[/latex]– are clockwise, while that created past[latex]\textbf{F}_{\textbf{B}}[/latex]– is counterclockwise.

Solution for (a)

Using the perpendicular lever artillery given in the figure, the second condition for equilibrium[latex]\boldsymbol{(\textbf{ internet }\tau=0)}[/latex]becomes

[latex]\boldsymbol{(0.350\textbf{ m})(55.0\textbf{ kg})(ix.80\textbf{ m/s}^ii)+(0.500\textbf{ yard})(thirty.0\textbf{ kg})(9.80\textbf{ m/southward}^ii)=(0.0800\textbf{ m})F_{\textbf{B}}}.[/latex]

Solving for[latex]\boldsymbol{F_{\textbf{B}}}[/latex]yields

[latex]\boldsymbol{F_{\textbf{B}}=iv.20\times10^3\textbf{ N}}.[/latex]

The ratio of the force the back muscles exert to the weight of the upper body plus its load is

[latex]\boldsymbol{\frac{F_{\textbf{B}}}{w_{\textbf{ub}}+w_{\textbf{box}}}}[/latex][latex]\boldsymbol{=}[/latex][latex]\boldsymbol{\frac{4200\textbf{ Northward}}{833\textbf{ N}}}[/latex][latex]\boldsymbol{=5.04.}[/latex]

This force is considerably larger than it would be if the load were not present.

Solution for (b)

More important in terms of its damage potential is the force on the vertebrae[latex]\boldsymbol{F}_{\textbf{Five}}.[/latex]The first condition for equilibrium[latex]\boldsymbol{(\textbf{ net }F=0)}[/latex]can be used to discover its magnitude and direction. Using[latex]\boldsymbol{y}[/latex]for vertical and[latex]\boldsymbol{10}[/latex]for horizontal, the condition for the net external forces along those axes to be zero

[latex]\boldsymbol{\textbf{ net }F_y=0\textbf{ and net }F_x=0.}[/latex]

Starting with the vertical ([latex]\boldsymbol{y}[/latex]) components, this yields

[latex]\boldsymbol{F_{\textbf{Vy}}-w_{\textbf{ub}}-w_{\textbf{box}}-F_{\textbf{B}}\sin\:29.0^0=0}.[/latex]

Thus,

[latex]\begin{array}{lcl} \boldsymbol{F_{\textbf{Vy}}} & \boldsymbol{=} & \boldsymbol{w_{\textbf{ub}}+w_{\textbf{box}}+F_{\textbf{B}}\:\sin\:29.0^0} \\ {} & \boldsymbol{=} & \boldsymbol{833\textbf{ Northward}+(4200\textbf{ N})\sin\:29.0^0} \end{array}[/latex]

yielding

[latex]\boldsymbol{F_{\textbf{Vy}}=ii.87\times10^3\textbf{ Northward}}.[/latex]

Similarly, for the horizontal ([latex]\boldsymbol{x}[/latex]) components,

[latex]\boldsymbol{F_{\textbf{Vx}}-F_{\textbf{B}}\:\cos\:29.0^0=0}[/latex]

yielding

[latex]\boldsymbol{F_{\textbf{Vx}}=3.67\times10^3\textbf{ N.}}[/latex]

The magnitude of[latex]\textbf{F}_{\textbf{V}}[/latex]is given by the Pythagorean theorem:

[latex]\boldsymbol{F_{\textbf{V}}=\sqrt{F_{\textbf{Vx}}^2+F_{\textbf{Vy}}^ii}=four.66\times10^3\textbf{ N.}}[/latex]

The direction of[latex]\textbf{F}_{\textbf{V}}[/latex]is

[latex]\boldsymbol{\theta=\tan^{-i}}[/latex][latex]\boldsymbol{(\frac{F_{\textbf{Vy}}}{F_{\textbf{Vx}}})}[/latex][latex]\boldsymbol{=38.0^0}.[/latex]

Note that the ratio of[latex]\boldsymbol{F_{\textbf{Five}}}[/latex]to the weight supported is

[latex]\boldsymbol{\frac{F_{\textbf{5}}}{w_{\textbf{ub}}+w_{\textbf{box}}}}[/latex][latex]\boldsymbol{=}[/latex][latex]\boldsymbol{\frac{4660\textbf{ N}}{833\textbf{ North}}}[/latex][latex]\boldsymbol{=5.59.}[/latex]

Discussion

This strength is about 5.six times greater than it would be if the person were continuing erect. The problem with the back is non so much that the forces are big—because like forces are created in our hips, knees, and ankles—but that our spines are relatively weak. Proper lifting, performed with the dorsum cock and using the legs to enhance the trunk and load, creates much smaller forces in the back—in this case, about 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 back, since these muscles have minor constructive perpendicular lever arms. The data shown here are analyzed in the preceding example, Instance 2.

What are the benefits of having almost skeletal muscles attached then close to joints? One advantage is speed because small musculus contractions can produce large movements of limbs in a short period of time. Other advantages are flexibility and agility, made possible past the large numbers of joints and the ranges over which they function. For example, information technology is difficult to imagine a arrangement with biceps muscles attached at the wrist that would be capable of the wide range of movement we vertebrates possess.

There are some interesting complexities in real systems of muscles, bones, and joints. For instance, the pin point in many joints changes location as the joint is flexed, then that the perpendicular lever arms and the mechanical reward of the organization change, besides. Thus the forcefulness the biceps musculus must exert to hold upwardly a book varies every bit the forearm is flexed. Similar mechanisms operate in the legs, which explicate, for case, why there is less leg strain when a bicycle seat is prepare at the proper peak. The methods employed in this department requite a reasonable description of real systems provided plenty is known nigh the dimensions of the system. There are many other interesting examples of forcefulness and torque in the body—a few of these are the subject of end-of-chapter bug.

Section Summary

• Statics plays an of import part in understanding everyday strains in our muscles and bones.
• Many lever systems in the torso take a mechanical advantage of significantly less than one, as many of our muscles are attached close to joints.
• Someone with practiced posture stands or sits in such as way that their heart of gravity lies directly in a higher place the pin betoken in their hips, thereby avoiding dorsum strain and damage to disks.

Conceptual Questions

1: Why are the forces exerted on the exterior world past the limbs of our bodies normally much smaller than the forces exerted by muscles inside the body?

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

3: Certain types of dinosaurs were bipedal (walked on 2 legs). What is a good reason that these creatures invariably had long tails if they had long necks?

4: Swimmers and athletes during competition demand to go through certain postures at the first of the race. Consider the residual of the person and why start-offs are so important for races.

5: If the maximum strength the biceps musculus can exert is 1000 Northward, tin we pick upwardly an object that weighs 1000 North? Explain your answer.

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

7: Explain 1 of the reasons why pregnant women often suffer from back strain late in their pregnancy.

Issues & Exercises

i: Verify that the force in the elbow joint in Instance 1 is 407 N, as stated in the text.

2: Two muscles in the back of the leg pull on the Achilles tendon as shown in Figure v. What total forcefulness do they exert?

Figure v. The Achilles tendon of the posterior leg serves to attach plantaris, gastrocnemius, and soleus muscles to calcaneus bone.

iii: The upper leg muscle (quadriceps) exerts a force of 1250 N, which is carried by a tendon over the kneecap (the patella) at the angles shown in Figure half-dozen. Discover the direction and magnitude of the strength exerted past the kneecap on the upper leg bone (the femur).

Figure 6. The knee joint works like a hinge to bend and straighten the lower leg. Information technology permits a person to sit down, stand, and pivot.

4: A device for exercising the upper leg muscle is shown in Effigy 7, together with a schematic representation of an equivalent lever system. Calculate the strength exerted by the upper leg musculus to lift the mass at a constant speed. Explicitly show how you lot follow the steps in the Problem-Solving Strategy for static equilibrium in Chapter 9.iv Applications of Statistics, Including Problem-Solving Strategies.

Figure 7. A mass is connected by pulleys and wires to the talocrural joint in this exercise device.

5: A person working at a drafting board may agree her head equally shown in Effigy 8, requiring muscle activeness to support the caput. The three major acting forces are shown. Calculate the direction and magnitude of the force supplied by the upper vertebrae[latex]\textbf{F}_{\textbf{V}}[/latex]to concur the head stationary, bold that this force acts along a line through the middle of mass equally do the weight and muscle force.

Figure 8.

six: We analyzed the biceps muscle case with the angle between forearm and upper arm set at[latex]\boldsymbol{90^0}.[/latex]Using the same numbers as in Example 1, find the strength exerted by the biceps muscle when the angle is[latex]\boldsymbol{120^0}[/latex]and the forearm is in a downwardly position.

7: Fifty-fifty when the caput is held erect, equally in Figure 9, its center of mass is non straight over the principal indicate of support (the atlanto-occipital articulation). The muscles at the back of the neck should therefore exert a force to keep the caput erect. That is why your head falls forrard when you autumn asleep in the class. (a) Calculate the strength exerted by these muscles using the information in the effigy. (b) What is the forcefulness exerted past the pivot on the head?

Figure 9. The heart of mass of the caput lies in front of its major point of support, requiring musculus action to hold the head erect. A simplified lever system is shown.

eight: A 75-kg man stands on his toes by 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 i foot? (b) Summate the strength at the pivot of the simplified lever organisation shown—that forcefulness is representative of forces in the ankle joint.

Figure 10. The muscles in the dorsum of the leg pull the Achilles tendon when one stands on one's toes. A simplified lever organisation is shown.

nine: A father lifts his child as shown in Effigy 11. What force should the upper leg musculus exert to lift the child at a constant speed?

Figure 11. A child existence lifted by a father's lower leg.

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

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

11: Integrated Concepts

Suppose nosotros replace the iv.0-kg book in Exercise 6 of the biceps muscle with an elastic do rope that obeys Hooke'due south Police. Presume its force constant[latex]\boldsymbol{g=600\textbf{ N/grand}}.[/latex](a) How much is the rope stretched (by equilibrium) to provide the same force[latex]\boldsymbol{F_{\textbf{B}}}[/latex]as in this instance? Assume the rope is held in the mitt at the same location as the book. (b) What force is on the biceps muscle if the practice rope is pulled direct up and then that the forearm makes an angle of[latex]\boldsymbol{25^0}[/latex]with the horizontal? Assume the biceps muscle is still perpendicular to the forearm.

12: (a) What force should the woman in Figure 13 exert on the flooring with each mitt to practise a push button-upwardly? Assume that she moves up at a constant speed. (b) The triceps muscle at the back of her upper arm has an effective lever arm of 1.75 cm, and she exerts force on the floor at a horizontal distance of 20.0 cm from the elbow joint. Calculate the magnitude of the force in each triceps muscle, and compare information technology to her weight. (c) How much piece of work does she exercise if her center of mass rises 0.240 m? (d) What is her useful ability output if she does 25 pushups in one minute?

Figure thirteen. A adult female doing pushups.

thirteen: Yous take just planted a sturdy ii-m-alpine palm tree in your front lawn for your female parent's birthday. Your brother kicks a 500 g brawl, which hits the top of the tree at a speed of 5 m/due south and stays in contact with it for 10 ms. The ball falls to the ground most 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 department of the root is merely xx cm. Furthermore, the soil around the roots is loose and we tin assume that an effective forcefulness is applied at the tip of the twenty cm length. What is the effective force exerted by the end of the tip of the root to go along the tree from toppling? Assume the tree will exist uprooted rather than curve. (c) What could you lot take done to ensure that the tree does not uproot hands?

fourteen: Unreasonable Results

Suppose two children are using a compatible seesaw that is 3.00 k long and has its heart of mass over the pivot. The first kid has a mass of 30.0 kg and sits 1.twoscore m from the pivot. (a) Calculate where the 2d eighteen.0 kg child must sit down to residuum the seesaw. (b) What is unreasonable about the result? (c) Which premise is unreasonable, or which premises are inconsistent?

15: Construct Your Ain Trouble

Consider a method for measuring the mass of a person's arm in anatomical studies. The discipline lies on her back, extends her relaxed arm to the side and ii scales are placed below the arm. One is placed under the elbow and the other under the back of her manus. Construct a problem in which yous calculate the mass of the arm and find its center of mass based on the scale readings and the distances of the scales from the shoulder articulation. You 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 data, if needed. You lot may wish to consult references to obtain reasonable mass values.

Solutions

Problems & Exercises

1:

[latex]\begin{array}{lcl} \boldsymbol{F_{\textbf{B}}} & \boldsymbol{=} & \boldsymbol{470\textbf{ North;}r_1=iv.00\textbf{ cm;}w_{\textbf{a}}=2.l\textbf{ kg;}r_2=16.0\textbf{ cm;}w_{\textbf{b}}=four.00\textbf{ kg;}r_3=38.0\textbf{ cm}} \\ \boldsymbol{F_{\textbf{E}}} & \boldsymbol{=} & \boldsymbol{w_{\textbf{a}}(\frac{r_2}{r_1}-ane)+w_{\textbf{b}}(\frac{r_3}{r_1}-1)} \\ {} & \boldsymbol{=} & \boldsymbol{(ii.50\textbf{ kg})(9.80\textbf{ m/due south}^ii)(\frac{16.0\textbf{ cm}}{4.0\textbf{ cm}}-1)} \\ {} & {} & \boldsymbol{+(iv.00\textbf{ kg})(ix.80\textbf{ m/south}^2)(\frac{38.0\textbf{ cm}}{4.00\textbf{ cm}}-1)} \\ {} & \boldsymbol{=} & \boldsymbol{407\textbf{ N}} \end{array}[/latex]

3:

[latex]\boldsymbol{1.1\times10^3\textbf{ N}}[/latex]

[latex]\boldsymbol{\theta=190^0\textbf{ ccw from positive }x\textbf{ axis}}[/latex]

5:

[latex]\boldsymbol{F_{\textbf{V}}=97\textbf{ Due north,}\:\theta=59^0}[/latex]

vii:

(a) 25 Due north downwards

(b) 75 N upwards

viii:

(a)[latex]\boldsymbol{F_{\textbf{A}}=ii.21\times10^three\textbf{ Northward}}[/latex]upward

(b)[latex]\boldsymbol{F_{\textbf{B}}=2.94\times10^3\textbf{ N}}[/latex]downward

x:

(a)[latex]\boldsymbol{F_{\textbf{teeth on bullet}}=1.2\times10^2\textbf{ Due north}}[/latex]upward

(b)[latex]\boldsymbol{F_{\textbf{J}}=84\textbf{ N}}[/latex]downwards

12:

(a) 147 N downwards

(b) 1680 N, iii.four times her weight

(c) 118 J

(d) 49.0 West

14:

a)[latex]\boldsymbol{\bar{x}_2=2.33\textbf{ m}}[/latex]

b) The seesaw is iii.0 one thousand long, and hence, at that place is only i.50 k of board on the other side of the pivot. The second kid is off the lath.

c) The position of the first kid must be shortened, i.e. brought closer to the pivot.

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Source: http://pressbooks-dev.oer.hawaii.edu/collegephysics/chapter/9-6-forces-and-torques-in-muscles-and-joints/

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