Two muscles in the back of the leg pull on the Achilles

Problem 1CQ What can you say about the velocity of a moving body that is in dynamic equilibrium? Draw a sketch of such a body using clearly labeled arrows to represent all external forces on the body.

Problem 1PE (a) When opening a door, you push on it perpendicularly with a force of 55.0 N at a distance of 0.850m from the hinges. What torque are you exerting relative to the hinges? (b) Does it matter if you push at the same height as the hinges?

Problem 1Q Describe several situations in which an object is not in equilibrium, even though the net force on it is zero.

Problem 3PE Two children push on opposite sides of a door during play. Both push horizontally and perpendicular to the door. One child pushes with a force of 17.5 N at a distance of 0.600 m from the hinges, and the second child pushes at a distance of 0.450 m. What force must the second child exert to keep the door from moving? Assume friction is negligible.

Problem 3Q You can find the center of gravity of a meter stick by resting it horizontally on your two index fingers, and then slowly drawing your fingers together. First the meter stick will slip on one finger, and then on the other, but eventually the fingers meet at the CG. Why does this work?

Problem 4PE Use the second condition for equilibrium (net ? = 0) to calculate Fp in Example 9.1, employing any data given or solved for in part (a) of the example. Example 9.1 She Saw Torques On A Seesaw The two children shown in ?Figure 9.9 ?are balanced on a seesaw of negligible mass. (This assumption is made to keep the example simple—more involved examples will follow.) The first child has a mass of 26.0 kg and sits 1.60 m from the pivot.(a) If the second child has a mass of 32.0 kg, how far is she from the pivot?

Your doctor's scale has arms on which weights slide to counter your weight, Fig. . These weights are much lighter than you are. How does this work? FIGURE 9-35 Question 4.

Problem 5PE Repeat the seesaw problem in Example 9.1 with the center of mass of the seesaw 0.160 m to the left of the pivot (on the side of the lighter child) and assuming a mass of 12.0 kg for the seesaw. The other data given in the example remain unchanged. Explicitly show how you follow the steps in the Problem-Solving Strategy for static equilibrium. Example 9.1 She Saw Torques On A Seesaw The two children shown in ?Figure 9.9 ?are balanced on a seesaw of negligible mass. (This assumption is made to keep the example simple—more involved examples will follow.) The first child has a mass of 26.0 kg and sits 1.60 m from the pivot.(a) If the second child has a mass of 32.0 kg, how far is she from the ?pivot? (b) What is ? p , the supporting force exerted by the pivot?

A ground retaining wall is shown in Fig. 9-36a. The ground, particularly when wet, can exert a significant force on the wall. (a) What force produces the torque to keep the wall upright? (b) Explain why the retaining wall in Fig. 9-36b would be much less likely to overturn than that in Fig. 9-36a. FIGURE 9-36 Question 5.

Problem 6PE Suppose a horse leans against a wall as in Figure 9.31. Calculate the force exerted on the wall assuming that force is horizontal while using the data in the schematic representation of the situation. Note that the force exerted on the wall is equal in magnitude and opposite in direction to the force exerted on the horse, keeping it in equilibrium. The total mass of the horse and rider is 500 kg. Take the data to be accurate to three digits.

Problem 7PE Two children of mass 20 kg and 30 kg sit balanced on a seesaw with the pivot point located at the center of the seesaw. If the children are separated by a distance of 3 m, at what distance from the pivot point is the small child sitting in order to maintain the balance?

Problem 7Q A ladder, leaning against a wall, makes a 60° angle with the ground. When is it more likely to slip: when a person stands on the ladder near the top or near the bottom? Explain.

A uniform meter stick supported at the mark is in equilibrium when a rock is suspended at the 0 -cm end (as shown in Fig. 9-37). Is the mass of the meter stick greater than, equal to, or less than the mass of the rock? Explain your reasoning. FIGURE 9-37 Question 8. FIGURE 9-37 Question 8.

Problem 8PE (a) Calculate the magnitude and direction of the force on each foot of the horse in Figure 9.31 (two are on the ground), assuming the center of mass of the horse is midway between the feet. The total mass of the horse and rider is 500kg. (b) What is the minimum coefficient of friction between the hooves and ground? Note that the force exerted by the wall is horizontal.

Problem 9Q Can the sum of the torques on an object be zero while the net force on the object is nonzero? Explain.

Problem 10PE A 17.0-m-high and 11.0-m-long wall under construction and its bracing are shown in Figure 9.32. The wall is in stable equilibrium without the bracing but can pivot at its base. Calculate the force exerted by each of the 10 braces if a strong wind exerts a horizontal force of 650 N on each square meter of the wall. Assume that the net force from the wind acts at a height halfway up the wall and that all braces exert equal forces parallel to their lengths. Neglect the thickness of the wall.

Figure 9-38 shows a cone. Explain how to lay it on a flat table so that it is in (a) stable equilibrium, (b) unstable equilibrium, ( ) neutral equilibrium. FIGURE 9-38 Question 10.

Problem 11PE (a) What force must be exerted by the wind to support a 2.50-kg chicken in the position shown in Figure 9.33? (b) What is the ratio of this force to the chicken’s weight? (c) Does this support the contention that the chicken has a relatively stable construction?

Which of the configurations of brick, or of Fig. 9-39, is the more likely to be stable? Why? FIGURE 9-39 Question 11. The dots indicate the CG of each brick. The fractions \(\frac{1}{4} \text { and } \frac{1}{2}\) indicate what portion of each brick is hanging beyond its support. Equation Transcription: Text Transcription: \frac{1}{4} and \frac{1}{2}

Problem 12PE Suppose the weight of the drawbridge in Figure 9.34 is supported entirely by its hinges and the opposite shore, so that its cables are slack. (a) What fraction of the weight is supported by the opposite shore if the point of support is directly beneath the cable attachments? (b) What is the direction and magnitude of the force the hinges exert on the bridge under these circumstances? The mass of the bridge is 2500 kg.

Why do you tend to lean backward when carrying a heavy load in your arms?

Suppose a 900-kg car is on the bridge in Figure 9.34 with its center of mass halfway between the hinges and the cable attachments. (The bridge is supported by the cables and hinges only.) (a) Find the force in the cables. (b) Find the direction and magnitude of the force exerted by the hinges on the bridge.

Place yourself facing the edge of an open door. Position your feet astride the door with your nose and abdomen touching the door's edge. Try to rise on your tiptoes. Why can't this be done?

Problem 14PE A sandwich board advertising sign is constructed as shown in Figure 9.35. The sign’s mass is 8.00 kg. (a) Calculate the tension in the chain assuming no friction between the legs and the sidewalk. (b) What force is exerted by each side on the hinge?

Why is it not possible to sit upright in a chair and rise to your feet without first leaning forward?

Problem 15PE (a) What minimum coefficient of friction is needed between the legs and the ground to keep the sign in Figure 9.35 in the position shown if the chain breaks? (b) What force is exerted by each side on the hinge?

Problem 15Q Why is it more difficult to do sit-ups when your knees are bent than when your legs are stretched out?

Problem 16PE A gymnast is attempting to perform splits. From the information given in Figure 9.36, calculate the magnitude and direction of the force exerted on each foot by the floor.

Name the type of equilibrium for each position of the ball in Fig. . FIGURE 9-40 Question 16.

To get up on the roof, a person (mass 70.0 kg) places a 6.00-m aluminum ladder (mass 10.0 kg) against the house on a concrete pad with the base of the ladder 2.00 m from the house. The ladder rests against a plastic rain gutter, which we can assume to be frictionless. The center of mass of the ladder is 2 m from the bottom. The person is standing 3 m from the bottom. What are the magnitudes of the forces on the ladder at the top and bottom?

Problem 17Q Is the Young's modulus for a bungee cord smaller or larger than that for an ordinary rope?

Problem 18PE In Figure 9.21, the cg of the pole held by the pole vaulter is 2.00 m from the left hand, and the hands are 0.700 m apart. Calculate the force exerted by (a) his right hand and (b) his left hand. (c) If each hand supports half the weight of the pole in Figure 9.19, show that the second condition for equilibrium (net ? = 0) is satisfied for a pivot other than the one located at the center of gravity of the pole. Explicitly show how you follow the steps in the Problem-Solving Strategy for static equilibrium described above.

Problem 18Q Examine how a pair of scissors or shears cuts through a piece of cardboard. Is the name "shears" justified? Explain.

Problem 19CQ Problem Explain one of the reasons why pregnant women often suffer from back strain late in their pregnancy.

Problem 19PE What is the mechanical advantage of a nail puller—similar to the one shown in Figure 9.23 —where you exert a force 45 cm from the pivot and the nail is 1.8 cm on the other side? What minimum force must you exert to apply a force of 1250 N to the nail?

Problem 19Q Materials such as ordinary concrete and stone are very weak under tension or shear. Would it be wise to use such a material for either of the supports of the cantilever shown in Fig. 9-9? If so, which one(s)? Explain.

Problem 20PE Suppose you needed to raise a 250-kg mower a distance of 6.0 cm above the ground to change a tire. If you had a 2.0-m long lever, where would you place the fulcrum if your force was limited to 300 N?

Problem 21PE a) What is the mechanical advantage of a wheelbarrow, such as the one in Figure 9.24, if the center of gravity of the wheelbarrow and its load has a perpendicular lever arm of 5.50 cm, while the hands have a perpendicular lever arm of 1.02 m? (b) What upward force should you exert to support the wheelbarrow and its load if their combined mass is 55.0 kg? (c) What force does the wheel exert on the ground?

Problem 22PE A typical car has an axle with 1.10 cm radius driving a tire with a radius of 27.5 cm . What is its mechanical advantage assuming the very simplified model in Figure 9.25(b)?

Problem 23PE What force does the nail puller in Exercise 9.19 exert on the supporting surface? The nail puller has a mass of 2.10 kg. Exercise 9.19: What is the mechanical advantage of a nail puller—similar to the one shown in Figure 9.23 —where you exert a force 45 cm from the pivot and the nail is 1.8 cm on the other side? What minimum force must you exert to apply a force of 1250 N to the nail?

Problem 24PE If you used an ideal pulley of the type shown in Figure 9.26(a) to support a car engine of mass 115 kg , (a) What would be the tension in the rope? (b) What force must the ceiling supply, assuming you pull straight down on the rope? Neglect the pulley system’s mass. Figure 9.26 ?(a) The combination of pulleys is used to multiply force. The force is an integral multiple of tension if the pulleys are frictionless. This pulley system has two cables attached to its load, thus applying a ? force of approximately 2?T . ? This machine has M ? A ? 2 .

Problem 26PE Problem Verify that the force in the elbow joint in Example 9.4 is 407 N, as stated in the text. Example 9.4: Muscles Exert Bigger Forces Than You Might Think Calculate the force the biceps muscle must exert to hold the forearm and its load as shown in ?Figure 9.27?, and compare this force with the weight of the forearm plus its load. You may take the data in the figure to be accurate to three significant figures.

Problem 25PE Repeat Exercise 9.24 for the pulley shown in Figure 9.26(c), assuming you pull straight up on the rope. The pulley system’s mass is 7.00 kg . Figure 9.26 ?(b) Three pulleys are used to lift a load in such a way that the mechanical advantage is about 3. Effectively, there are three cables attached to the load. Exercise 9.24: If you used an ideal pulley of the type shown in Figure 9.26(a) to support a car engine of mass 115 kg , (a) What would be the tension in the rope? (b) What force must the ceiling supply, assuming you pull straight down on the rope? Neglect the pulley system’s mass. Figure 9.26 ?(a) The combination of pulleys is used to multiply force. The force is an integral multiple of tension if the pulleys are frictionless. This pulley system has two cables a?ttached to its load, thus applying a force of approximately 2?T . ? This machine has M ? A ? 2 .

Problem 27PE Two muscles in the back of the leg pull on the Achilles tendon as shown in Figure 9.37. What total force do they exert?

Problem 28PE 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 9.38. Find the direction and magnitude of the force exerted by the kneecap on the upper leg bone (the femur).

Problem 29PE A device for exercising the upper leg muscle is shown in Figure 9.39, together with a schematic representation of an equivalent lever system. Calculate the force exerted by the upper leg muscle to lift the mass at a constant speed. Explicitly show how you follow the steps in the ProblemSolving Strategy for static equilibrium in Applications of Statistics, Including Problem-Solving Strategies.

Problem 30PE A person working at a drafting board may hold her head as shown in Figure 9.40, requiring muscle action to support the head. The three major acting forces are shown. Calculate the direction and magnitude of the force supplied by the upper vertebrae FV to hold the head stationary, assuming that this force acts along a line through the center of mass as do the weight and muscle force.

Problem 31PE We analyzed the biceps muscle example with the angle between forearm and upper arm set at 90º. Using the same numbers as in Example 9.4, find the force exerted by the biceps muscle when the angle is 120º and the forearm is in a downward position. Example 9.4: Muscles Exert Bigger Forces Than You Might Think Calculate the force the biceps muscle must exert to hold the forearm and its load as shown in ?Figure 9.27?, and compare this force with the weight of the forearm plus its load. You may take the data in the figure to be accurate to three significant figures.

Problem 32PE Even when the head is held erect, as in Figure 9.41, its center of mass is not directly over the principal point of support (the atlanto-occipital joint). The muscles at the back of the neck should therefore exert a force to keep the head erect. That is why your head falls forward when you fall asleep in the class. (a) Calculate the force exerted by these muscles using the information in the figure. (b) What is the force exerted by the pivot on the head?

Problem 33PE A 75-kg man stands on his toes by exerting an upward force through the Achilles tendon, as in Figure 9.42. (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 ankle joint.

Problem 34PE A father lifts his child as shown in Figure 9.43. What force should the upper leg muscle exert to lift the child at a constant speed?

Unlike most of the other muscles in our bodies, the masseter muscle in the jaw, as illustrated in Figure 9.44, is attached relatively far from the joint, enabling large forces to be exerted by the back teeth. (a) Using the information in the figure, calculate the force exerted by the lower teeth on the bullet. (b) Calculate the force on the joint.

Problem 36PE Integrated Concepts Suppose we replace the 4.0-kg book in Exercise 9.31 of the biceps muscle with an elastic exercise rope that obeys Hooke’s Law. Assume its force constant k = 600 N/m . (a) How much is the rope stretched (past equilibrium) to provide the sam? e force ?FB as in this example? Assume the rope is held in the hand at the same location as the book. (b) What force is on the biceps muscle if the exercise rope is pulled straight up so that the forearm makes an angle of 25º with the horizontal? Assume the biceps muscle is still perpendicular to the forearm. Reference Exercise 9.31: We analyzed the biceps muscle example with the angle between forearm and upper arm set at 90º. Using the same numbers as in Example 9.4, find the force exerted by the biceps muscle when the angle is 120º and the forearm is in a downward position. Example 9.4: Muscles Exert Bigger Forces Than You Might Think Calculate the force the biceps muscle must exert to hold the forearm and its load as shown in ?Figure 9.27?, and compare this force with the weight of the forearm plus its load. You may take the data in the figure to be accurate to three significant figures.

(III) Four bricks are to be stacked at the edge of a table, each brick overhanging the one below it, so that the top brick extends as far as possible beyond the edge of the table. To achieve this, show that successive bricks must extend no more than (starting at the top) \(\frac{1}{2}, \frac{1}{4}=\frac{1}{6}, a n d \frac{1}{8}\) of their length beyond the one below (Fig. Is the top brick completely beyond the base? ( ) Determine a general formula for the maximum total distance spanned by bricks if they are to remain stable. A builder wants to construct a corbeled arch (Fig. based on the principle of stability discussed in and above. What minimum number of bricks, each long, is needed if the arch is to span ? FIGURE 9-67 Problem 37. Equation Transcription: Text Transcription: \frac{1}{2}, \frac{1}{4}=\frac{1}{6}, a n d \frac{1}{8}

Problem 37PE (a) What force should the woman in Figure 9.45 exert on the floor with each hand to do a push-up? 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 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?

Problem 38PE You have just planted a sturdy 2-m-tall palm tree in your front lawn for your mother’s birthday. Your brother kicks a 500 g ball, which hits the top of the tree at a speed of 5 m/s and stays in contact with it for 10 ms. The ball falls to the ground 342 Chapter 9 | Statics and Torque 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 can assume that an effective force is applied 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 will be uprooted rather than bend. (c) What could you have done to ensure that the tree does not uproot easily?

Problem 39PE Unreasonable Results Suppose two children are using a uniform seesaw that is 3.00 m long and has its center of mass over the pivot. The first child has a mass of 30.0 kg and sits 1.40 m from the pivot. (a) Calculate where the second 18.0 kg child must sit to balance the seesaw. (b) What is unreasonable about the result? (c) Which premise is unreasonable, or which premises are inconsistent?

(I) A nylon string on a tennis racket is under a tension of 275 N. If its diameter is 1.00 mm, by how much is it lengthened from its untensioned length of 30.0 cm?

Problem 42P (II) One liter of alcohol (1000 cm3) in a flexible container is carried to the bottom of the sea, where the pressure is 2.6 x 106 N/m2. What will be its volume there?

Problem 40PE Construct Your Own Problem Consider a method for measuring the mass of a person’s arm in anatomical studies. The subject 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 back of her hand. Construct a problem in which you 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 joint. You must include a free body diagram of the arm to direct the analysis. Consider changing the position of the scale under the hand to provide more information, if needed. You may wish to consult references to obtain reasonable mass values.

Problem 43P (II) A 15-cm-long tendon was found to stretch 3.7 mm by a force of 13.4 N. The tendon was approximately round with an average diameter of 3.5 mm. Calculate Young's modulus of this tendon.

Problem 44P (II) How much pressure is needed to compress the volume of an iron block by 0.10%? Express your answer in N/m2, and compare it to atmospheric pressure (1.0 x 105 N/m2).

Problem 45P (II) At depths of 2000 m in the sea, the pressure is about 200 times atmospheric pressure (1 atm = 1.0 x 105 N/m2). By what percentage does the interior space of an iron bathysphere's volume change at this depth?

Problem 46P (III) A scallop forces open its shell with an elastic material called abductin, whose Young's modulus is about 2.0 x 106 N/m2. If this piece of abductin is 3.0 mm thick and has a cross-sectional area of 0.50 cm2, how much potential energy does it store when compressed 1.0 mm?

Problem 48P (I) The femur bone in the human leg has a minimum effective cross section of about 3.0 cm2 (= 3.0 x 10-4 m2). How much compressive force can it withstand before breaking?

Problem 49P (II) (a) What is the maximum tension possible in a 1.00-mm-diameter nylon tennis racket string? (b) If you want tighter strings, what do you do to prevent breakage: use thinner or thicker strings? Why? What causes strings to break when they are hit by the ball?

Problem 55P (II) How high must a pointed arch be if it is to span a space 8.0 m wide and exert one-third the horizontal force at its base that a round arch would?

(II) The subterranean tension ring that exerts the balancing horizontal force on the abutments for the dome in Fig. 9-34 is 36-sided, so each segment makes a \(10^{\circ}\) angle with the adjacent one (Fig. 9-70). Calculate the tension F that must exist in each segment so that the required force of \(4.2 \times 10^5 \mathrm{~N}\) can be exerted at each corner (Example 9-13).

The center of gravity of a loaded truck depends on how the truck is packed. If it is high and wide, and its CG is above the ground, how steep a slope be parked on without tipping over (Fig. 9-75)? FIGURE 9-75 Problem 63.

When a mass of is hung from the middle of a fixed straight aluminum wire, the wire sags to make an angle of \(12^{\circ}\) with the horizontal as shown in Fig. . Determine the radius of the wire. FIGURE 9-77 Problem 65. Equation Transcription: Text Transcription: 12°

A cube of side rests on a rough floor. It is subjected to a steady horizontal pull , exerted a distance above the floor as shown in Fig. . As is increased, the block will either begin to slide, or begin to tip over. Determine the coefficient of static friction \(\mu_{s}\) so that the block begins to slide rather than tip; (b) the block begins to tip. [Hint: Where will the normal force on the block act if it tips?] FIGURE 9-81 Problem 69. Equation Transcription: Text Transcription: \mu_s

Problem 80GP There is a maximum height of a uniform vertical column made of any material that can support itself without buckling, and it is independent of the cross-sectional area (why?). Calculate this height for (a) steel (density ), and (b) granite (density 2.7 X 103 kg/m3).

A woman holds a -m-long uniform pole as shown in Fig. ( ) Determine the forces she must exert with each hand (magnitude and direction). To what position should she move her left hand so that neither hand has to exert a force greater than (c) ? FIGURE 9-83 Problem 71.

Problem 2Q A bungee jumper momentarily comes to rest at the bottom of the dive before he springs back upward. At that moment, is the bungee jumper in equilibrium? Explain.

(II) Find the tension in the two cords shown in Fig. Neglect the mass of the cords, and assume that the angle \(\theta \text { is } 33^{\circ}\) and the mass is . FIGURE 9-45 Problem 11. Equation Transcription: Text Transcription: \theta is 33^\circ

Problem 11CQ Two ice skaters, Megan and Jason, push off from each other on frictionless ice. Jason’s mass is twice that of Megan. a. Which skater, if either, experiences the greater impulse during the push? Explain. b. Which skater, if either, has the greater speed after the pushoff? Explain.

Problem 10P (II) Calculate FA and FB for the uniform cantilever shown in Fig. 9–9 whose mass is 1200 kg.

Problem 7CQ Two particles collide, one of which was initially moving and the other initially at rest. a. Is it possible for ?both particles to be at rest after the collision? Give an example in which this happens, or explain why it can’t happen. b. Is it possible for ?one particle to be at rest after the collision? Give an example in which this happens, or explain why it can’t happen.

(II) Calculate the forces \(F_{A} \text { and } F_{B}\) that the supports exert on the diving board of Fig. 9-42 when a person stands at its tip. (a) Ignore the weight of the board. (b) Take into account the board's mass of . Assume the board's is at its center. Equation Transcription: Text Transcription: F_A and F_B

Problme 6CQ Two students stand at rest, facing each other on frictionless skates. They then start tossing a heavy ball back and forth between them. Describe their subsequent motion.

(I) Calculate the mass needed in order to suspend the leg shown in Fig. Assume the leg (with cast) has a mass of , and its is from the hip joint; the sling is from the hip joint.

(I) How far out on the diving board (Fig. 9-42) would a \(58-\mathrm{kg}\) diver have to be to exert a torque of \(1100 \mathrm{~m} \cdot \mathrm{N}\) on the board, relative to the left (A) support post?

3CQ A 0.2 kg plastic cart and a 20 kg lead cart can roll without friction on a horizontal surface. Equal forces are used to push both carts forward for a time of 1 s, starting from rest. After the force is removed at t = 1 s, is the momentum of the plastic cart greater than, less than, or equal to the momentum of the lead cart? Explain.

(II) A uniform steel beam has a mass of . On it is resting half of an identical beam, as shown in Fig. . What is the vertical support force at each end?

(II) Figure shows a pair of forceps used to hold a thin plastic rod firmly. If each finger squeezes with a force \(F_{T}=F_{B}=11.0 \mathrm{~N}\), what force do the forceps jaws exert on the plastic rod? Equation Transcription: Text Transcription: FT=FB=11.0 N

(II) Calculate the tension \(F_{T}\) in the wire that supports the beam shown in Fig. , and the force \(\vec{F}_{W}\) exerted by the wall on the beam (give magnitude and direction). Equation Transcription: Text Transcription: F_{T} \ \vec{F}_{W}

(II) A -tall person lies on a light (massless) board which is supported by two scales, one under the top of her head and one beneath the bottom of her feet (Fig. ). The two scales read, respectively, and . What distance is the center of gravity of this person from the bottom of her feet?

(II) A shop sign weighing is supported by a uniform beam as shown in Fig. 9-54. Find the tension in the guy wire and the horizontal and vertical forces exerted by the hinge on the beam.

(II) A traffic light hangs from a pole as shown in Fig. . The uniform aluminum pole is long and has a mass of . The mass of the traffic light is . Determine the tension in the horizontal massless cable , and the vertical and horizontal components of the force exerted by the pivot on the aluminum pole.

(II) The \(72-\mathrm{kg}\)-man's hands in Fig. 9-56 are \(36 \mathrm{~cm}\) apart. His CG is located 75% of the distance from his right hand toward his left. Find the force on each hand due to the ground.

(II) The two trees in Fig. 9-58 are \(7.6 \mathrm{~m}\) apart. A backpacker is trying to lift his pack out of the reach of bears. Calculate the magnitude of the force \(\overrightarrow{\mathbf{F}}\) that he must exert downward to hold a 19-kg backpack so that the rope sags at its midpoint by (a) \(1.5 \mathrm{~m}\), (b) \(0.15 \mathrm{~m}\)

(II) A uniform meter stick with a mass of \(180 \mathrm{~g}\) is supported horizontally by two vertical strings, one at the \(0-\mathrm{cm}\) mark and the other at the 90-cm mark (Fig. 9-57). What is the tension in the string (a) at \(0 \mathrm{~cm}\)? (b) at \(90 \mathrm{~cm}\)?

(III) A door high and wide has a mass of . A hinge from the top and another hinge from the bottom each support half the door's weight (Fig. 9-59). Assume that the center of gravity is at the geometrical center of the door, and determine the horizontal and vertical force components exerted by each hinge on the door.

(III) Consider a ladder with a painter climbing up it (Fig. 9-61). If the mass of the ladder is , the mass of the painter is , and the ladder begins to slip at its base when her feet are of the way up the length of the ladder, what is the coefficient of static friction between the ladder and the floor? Assume the wall is frictionless.

(III) A uniform ladder of mass and length leans at an angle \(\theta\) against a frictionless wall, Fig. cient of static the ladder and determine a minimum angle ladder will not Equation Transcription: Text Transcription: \theta

(III) A person wants to push a lamp (mass ) across the floor, for which the coefficient of friction is . Calculate the maximum height above the floor at which the person can push the lamp so that it slides rather than tips (Fig. 9-62).

(III) Two wires run from the top of a pole tall that supports a volleyball net. The two wires are anchored to the ground apart, and each is from the pole (Fig. 9-63). The tension in each wire is . What is the tension in the net, assumed horizontal and attached at the top of the pole?

Problem 31P Approximately what magnitude force, FM, must the extensor muscle in the upper arm exert on the lower arm to hold a 7.3-kg shot put (Fig. 9–64)? Assume the lower arm has a mass of 2.8 kg and its CG is 12 cm from the elbow-joint pivot. FIGURE 9–64

(I) Suppose the point of insertion of the biceps muscle into the lower arm shown in Fig. 9-13a (Example 9-8) is instead of ; how much mass could the person hold with a muscle exertion of ?

(II) (a) Calculate the force, \(F_{M}\), required of the "deltoid" muscle to hold up the outstretched arm shown in Fig. . The total mass of the arm is . (b) Calculate the magnitude of the force \(F_{J}\) exerted by the shoulder joint on the upper arm. Equation Transcription: Text Transcription: F_{M} F_{J}

(II) Redo Example 9-9, assuming now that the person is less bent over so that the \(30^{\circ}\) in Fig. is instead \(45^{\circ}\). What will be the magnitude of \(F_{V}\) on the vertebra? Equation Transcription: Text Transcription: 30^\circ 45^\circ F_V

(II) The Achilles tendon is attached to the rear of the foot as shown in Fig. . When a person elevates himself just barely off the floor on the "ball of one foot," estimate the tension \(F_{T}) in the Achilles tendon (pulling upward), and the (downward) force \(F_{B}\) exerted by the lower leg bone on the foot. Assume the person has a mass of and is twice as long as . Equation Transcription: Text Transcription: F_T F_B

(II) Suppose the hand in Problem 32 holds a \(15-\mathrm{kg}\) mass. What force, \(F_{\mathrm{M}}\), is required of the deltoid muscle, assuming the mass is \(52 \mathrm{~cm}\) from the shoulder joint?

The mobile in Fig. is in equilibrium. Object has mass of Determine the masses of objects , and D. (Neglect the weights of the crossbars.)

What minimum horizontal force is needed to pull a wheel of radius and mass over a step of height as shown in Fig. \(9-72(R>h)\)? (a) Assume the force is applied at the top edge as shown. (b) Assume the force is applied instead at the wheel's center. Equation Transcription: Text Transcription: 9-72(R>h)

Problem 58GP A tightly stretched “high wire” is 46 m long. It sags 2.2 m when a 60.0-kg tightrope walker stands at its center. What is the tension in the wire? Is it possible to increase the tension in the wire so that there is no sag?

Problem 60GP A 25-kg round table is supported by three legs equal distances apart on the edge. What minimum mass, placed on the table’s edge, will cause the table to overturn?

(I) Three forces are applied to a tree sapling, as shown in Fig. 9-41, to stabilize it. If \(\overrightarrow{\mathbf{F}}_{\mathrm{A}}=310 \mathrm{~N}\) and \(\overrightarrow{\mathbf{F}}_{\mathrm{B}}=425 \mathrm{~N}\), find \(\overrightarrow{\mathbf{F}}_{\mathrm{C}}\) in magnitude and direction.

Two cords support a chandelier in the manner shown in Fig. 9–4 except that the upper wire makes an angle of \(45^{\circ}\) with the ceiling. If the cords can sustain a force of 1550 N without breaking, what is the maximum chandelier weight that can be supported? Equation Transcription: Text Transcription: 45^\circ

Problem 8P A 140-kg horizontal beam is supported at each end. A 320-kg piano rests a quarter of the way from one end. What is the vertical force on each of the supports?

(II) A 75-kg adult sits at one end of a 9.0-m-long board. His 25-kg child sits on the other end. (a) inhere should the pivot be placed so that the board is balanced, ignoring the board’s mass? (b) Fine the pivot point if the board is uniform and has a mass of 15 kg.

(II) Find the tension in the two wires supporting the traffic light shown in Fig. 9-53.

(II) How close to the edge of the 20.0-kg table shown in Fig. 9–47 can a 66.0-kg person sit without tipping it over?

A 0.60-kg sheet hangs from a massless clothesline as shown in Fig. 9–48. The clothesline on either side of the sheet makes an angle of \(3.5^{\circ}\) with the horizontal. Calculate the tension in the clothesline on either side of the sheet. Why is the tension so much greater than the weight of the sheet? Equation Transcription: Text Transcription: 3.5°

Calculate \(F_{A} \text { and } F_{B}\) for the beam shown in Fig. 9–49. The downward forces represent the weights of machinery on the beam. Assume the beam is uniform and has a mass of 250 kg. Equation Transcription: Text Transcription: F_A and F_B

Three children are trying to balance on a seesaw, which consists of a fulcrum rock, acting as a pivot at the center, and a very light board 3.6 m long (Fig. 9–50). Two playmates are already on either end. Boy A has a mass of 50 kg, and girl B a mass of 35 kg. Where should girl C, whose mass is 25 kg, place herself so as to balance the seesaw?

Explain why touching your toes while you are seated on the floor with outstretched legs produces less stress on the lower spinal column than when touching your toes from a standing position. Use a diagram.

(I) Calculate the torque about the front support post (B) of a diving board, Fig. 9-42, exerted by a person from that post.

Problem 36P The Leaning Tower of Pisa is 55 m tall and about 7.0 m in diameter. The top is 4.5 m off center. Is the tower in stable equilibrium? If so, how much farther can it lean before it becomes unstable? Assume the tower is of uniform composition.

(I) A marble column of cross-sectional area \(1.2 \mathrm{~m}^2\) supports a mass of \(25,000 \mathrm{~kg}\). (a) What is the stress within the column? (b) What is the strain?

Problem 40P By how much is the column in Problem 39 shortened if it is 9.6 m high? Problem 39 A marble column of cross-sectional area 1.2 m2 supports a mass of 25,000 kg. (a) What is the stress within the column? (b) What is the strain?

(I) A sign (mass 2100 kg) hangs from the end of a vertical steel girder with a cross-sectional area of 0.15 \(m^2\). (a) What is the stress within the girder? (b) What is the strain on the girder? (c) If the girder is 9.50 m long, how much is it lengthened? (Ignore the mass of the girder itself.)

(III) A pole projects horizontally from the front wall of a shop. A 5.1-kg sign hangs from the pole at a point 2.2 m from the wall (Fig. 9-68). (a) What is the torque due to this sign calculated about the point where the pole meets the wall? (b) If the pole is not to fall off, there must be another torque exerted to balance it. What exerts this torque? Use a diagram to show how this torque must act. (c) Discuss whether compression, tension, and/or shear play a role in part (b).

(II) If a compressive force of \(3.6 \times 10^{4} \ \mathrm {N}\) is exerted on the end of a 22-cm-long bone of cross-sectional area \(3.6 \ \mathrm {cm}^2\), (a) will the bone break, and (b) if not, by how much does it shorten?

Problem 51P (a) What is the minimum cross-sectional area required of a vertical steel cable from which is suspended a 320-kg chandelier? Assume a safety factor of 7.0 (b) If the cable is 7.5 m long, how much does it elongate?

(II) Assume the supports of the uniform cantilever shown in Fig. (mass ) are made of wood. Calculate the minimum cross-sectional area required of each, assuming a safety factor of .

Problem 53P An iron bolt is used to connect two iron plates together. The bolt must withstand shear forces up to about 3200 N. Calculate the minimum diameter for the bolt, based on a safety factor of 6.0.

Problem 54P A steel cable is to support an elevator whose total (loaded) mass is not to exceed 3100 kg. If the maximum acceleration of the elevator is 1.2 m/s2, calculate the diameter of cable required. Assume a safety factor of 7.0.

When a wood shelf of mass is fastened inside a slot in a vertical support as shown in Fig. , the support exerts a torque on the shelf. (a) Draw a free-body diagram for the shelf, assuming three vertical forces (two exerted by the support slot-explain why). Then calculate (b) the magnitudes of the three forces and the torque exerted by the support (about the left end of the shelf).

A 50 -story building is being planned. It is to be high with a base by . Its total mass will be about \(1.8 \times 10^{7} \mathrm{~kg}\), and its weight therefore about \(1.8 \times 10^{8} \mathrm{~N}\). Suppose a \(200-\mathrm{km} / \mathrm{h}\) wind exerts a force of \(950 \mathrm{~N} / \mathrm{m}^{2}\) over the -wide face (Fig. . Calculate the torque about the potential pivot point, the rear edge of the building (where \(\vec{F}_{E}\) acts in Fig. 9-74), and determine whether the building will topple. Assume the total force of the wind acts at the midpoint of the building's face, and that the building is not anchored in bedrock. [Hint: \(\vec{F}_{E}\) in Fig. represents the force that the Earth would exert on the building in the case where the building would just begin to tip.] Equation Transcription: Text Transcription: 1.8 \times 10^7 kg 1.8 \times 10^8 N 200 - km / h 950 N / m^2 \vec F_E \vec F_E

In Fig. 9-76, consider the right-hand (northernmost) section of the Golden Gate Bridge, which has a length \(d_{1}=343 \mathrm{~m}\).. Assume the of this span is halfway between the tower and anchor. Determine \(F_{T 1} \text { and } F_{T 2}\) (which act on the northernmost cable) in terms of , the weight of the northernmost span, and calculate the tower height needed for equilibrium. Assume the roadway is supported only by the suspension cables, and neglect the mass of the cables and vertical wires. [Hint:\(F_{T 3}\) does not act on this section.] Equation Transcription: Text Transcription: d_1=343 m F_T1 and F_T2 F_T3

The forces acting on a aircraft flying at constant velocity are shown in Fig. 9-78. The engine thrust, \(F_{T}=5.0 \times 10^{5} N\), acts on a line below the . Determine the drag force \(F_{D}\) and the distance above the CM that it acts. Assume \(\vec{F}_{D} \text { and } \vec{F}_{T}\) are horizontal. Equation Transcription: Text Transcription: F_T=5.0 \times 10^5 N F_D \vec F_D and \vec F_T

A uniform flexible steel cable of weight is suspended between two points at the same elevation as shown in Fig. , where \(\theta=60^{\circ}\). Determine the tension in the cable at its lowest point, and at the points of attachment. (c) What is the direction of the tension force in each case? Equation Transcription: Text Transcription: \theta=60^\circ

A 20.0-m-long uniform beam weighing \(550 \mathrm{~N}\) rests on walls \(\mathrm{A}\) and \(\mathrm{B}\), as shown in Fig. 9-80. (a) Find the maximum weight of a person who can walk to the extreme end D without tipping the beam. Find the forces that the walls \(\mathrm{A}\) and \(\mathrm{B}\) exert on the beam when the person is standing: (b) at D; (c) at a point \(2.0 \mathrm{~m}\) to the right of B; (d) \(2.0 \mathrm{~m}\) to the right of A.

A painter is on a uniform scaffold supported from above by ropes (Fig. There is a pail of paint to one side, as shown. Can the painter walk safely to both ends of the scaffold? If not, which end(s) is dangerous, and how close to the end can he approach safely?

A man doing push-ups pauses in the position shown in Fig. His mass . Determine the normal force exerted by the floor on each hand; on each foot.

A 20-kg sphere rests between two smooth planes as shown in Fig. 9-85. Determine the magnitude of the force acting on the sphere exerted by each plane.

A trailer is attached to a stationary truck at point B, Fig. 9-86. Determine the normal force exerted by the road on the rear tires at , and the vertical force exerted on the trailer by the support .

Problem 75GP Parachutists whose chutes have failed to open have been known to survive if they land in deep snow. Assume that a 75-kg parachutist hits the ground with an area of impact of 0.30 m2 at a velocity of 60 m/s, and that the ultimate strength of body tissue is 5 × 105 N/m2. Assume that the person is brought to rest in 1.0 m of snow. Show that the person may escape serious injury.

A steel wire 2.0 mm in diameter stretches by 0.030% when a mass is suspended from it. How large is the mass?

In Example in Chapter 7, we calculated the impulse and average force on the leg of a person who jumps down to the ground. If the legs are not bent upon landing, so that the body moves a distance of only during collision, determine (a) the stress in the tibia (a lower leg bone of area = \(3.0 \times 10^{-4} \mathrm{~m}^{2}\)) and (b) whether or not the bone will break. (c) Repeat for a bent-knees landing \((d=50.0 \mathrm{~cm})\). Equation Transcription: Text Transcription: 3.0 x 10^-4 m^2 (d=50.0 cm)

Problem 78GP The roof over a 7.0-m × 10.0-m room in a school has a total mass of 12,600 kg. Tire roof is to be supported by vertical “2 × 4s” (actually about 4.0 cm × 9.0 cm) along the 10.0-m sides. How many supports are required on each side, and how far apart must they be? Consider only compression, and assume a safety factor of 12.

A object is being lifted by pulling on the ends of a 1.00-mm-diameter nylon string that goes over two -high poles that are apart, as shown in Fig. . How high above the floor will the object be when the string breaks?