Flying Buttress. (a) A symmetric building has a roof

Problem 5DQ In Section 11.2 we always assumed that the value of g was the same at all points on the body. This is ?not a good approximation if the dimensions of the body are great enough, because the value of g decreases with altitude. If this is taken into account, will the center of gravity of a long, vertical rod be above, below, or at its center of mass? Explain how this can be used to keep the long axis of an orbiting spacecraft pointed toward the earth. (This would be useful for a weather satellite that must always keep its camera lens trained on the earth.) The moon is not exactly spherical but is somewhat elongated. Explain why this same effect is responsible for keeping the same face of the moon pointed toward the earth at all times.

Problem 6E Two people are carrying a uniform wooden board that is 3.00 m long and weighs 160 N. If one person applies an upward force equal to 60 N at one end, at what point does the other person lift? Begin with a free-body diagram of the board.

Problem 2DQ (a) Is it possible for an object to be in translational equilibrium (the first condition) but not in rotational equilibrium (the second condition)? Illustrate your answer with a simple example. (b) Can an object be in rotational equilibrium yet not in translational equilibrium? Justify your answer with a simple example.

Problem 2E The center of gravity of a 5.00-kg irregular object is shown in ?Fig. E11.2. You need to move the center of gravity 2.20 cm to the left by gluing on a 1.50-kg mass, which will then be considered as part of the object. Where should the center of gravity of this additional mass be located?

Problem 3DQ Car tires are sometimes “balanced” on a machine that pivots the tire and wheel about the center. Weights are placed around the wheel rim until it does not tip from the horizontal plane. Discuss this procedure in terms of the center of gravity.

Problem 3E A uniform rod is 2.00 m long and has mass 1.80 kg. A 2.40-kg clamp is attached to the rod. How far should the center of gravity of the clamp be from the left-hand end of the rod in order for the center of gravity of the composite object to be 1.20 m from the left-hand end of the rod?

You are balancing a wrench by suspending it at a single point. Is the equilibrium stable, unstable, or neutral if the point is above, at, or below the wrench’s center of gravity? In each case give the reasoning behind your answer. (For rotation, a rigid body is in stable equilibrium if a small rotation of the body produces a torque that tends to return the body to equilibrium; it is in unstable equilibrium if a small rotation produces a torque that tends to take the body farther from equilibrium; and it is in neutral equilibrium if a small rotation produces no torque.)

Problem 7DQ You can probably stand flatfooted on the floor and then rise up and balance on your tiptoes. Why are you unable do it if your toes are touching the wall of your room? (Try it!)

Problem 8DQ You freely pivot a horseshoe from a horizontal nail through one of its nail holes. You then hang a long string with a weight at its bottom from the same nail, so that the string hangs vertically in front of the horseshoe without touching it. How do you know that the horseshoe’s center of gravity is along the line behind the string? How can you locate the center of gravity by repeating the process at another nail hole? Will the center of gravity be within the solid material of the horseshoe?

Problem 7 Two people carry a heavy electric motor by placing it on a light board 2.00 m long. One person lifts at one end with a force of 400 N, and the other lifts the opposite end with a force of 600 N. (a) What is the weight of the motor, and where along the board is its center of gravity located? (b) Suppose the board is not light but weighs 200 N, with its center of gravity at its center, and the two people each exert the same forces as before. What is the weight of the motor in this case, and where is its center of gravity located?

Problem 8E A 60.0-cm, uniform, 50.0-N shelf is supported horizontally by two vertical wires attached to the sloping ceiling (?Fig. E11.8?). A very small 25.0-N tool is placed on the shelf midway between the points where the wires are attached to it. Find the tension in each wire. Begin by making a free-body diagram of the shelf.

Problem 9DQ An object consists of a ball of weight W glued to the end of a uniform bar also of weight W. If you release it from rest, with the bar horizontal, what will its behavior be as it falls if air resistance is negligible? Will it (a) remain horizontal; (b) rotate about its center of gravity; (c) rotate about the ball; or (d) rotate so that the ball swings downward? Explain your reasoning.

Problem 9E A 350-N, uniform, 1.50-m bar is suspended horizontally by two vertical cables at each end. Cable A can support a maximum tension of 500.0 N without breaking, and cable B can support up to 400.0 N. You want to place a small weight on this bar. (a) What is the heaviest weight you can put on without breaking either cable, and (b) where should you put this weight?

Problem 10DQ Suppose that the object in Question 11.9 is released from rest with the bar tilted at 60° above the horizontal with the ball at the upper end. As it is falling, will it (a) rotate about its center of gravity until it is horizontal; (b) rotate about its center of gravity until it is vertical with the ball at the bottom; (c) rotate about the ball until it is vertical with the ball at the bottom; or (d) remain at 60° above the horizontal? Q11.9 An object consists of a ball of weight W glued to the end of a uniform bar also of weight W. If you release it from rest, with the bar horizontal, what will its behavior be as it falls if air resistance is negligible? Will it (a) remain horizontal; (b) rotate about its center of gravity; (c) rotate about the ball; or (d) rotate so that the ball swings downward? Explain your reasoning.

Problem 10E A uniform ladder 5.0 m long rests against a frictionless, vertical wall with its lower end 3.0 m from the wall. The ladder weighs 160 N. The coefficient of static friction between the foot of the ladder and the ground is 0.40. A man weighing 740 N climbs slowly up the ladder. Start by drawing a free-body diagram of the ladder. (a) What is the maximum friction force that the ground can exert on the ladder at its lower end? (b) What is the actual friction force when the man has climbed 1.0 m along the ladder? (c) How far along the ladder can the man climb before the ladder starts to slip?

Problem 11DQ Why must a water skier moving with constant velocity lean backward? What determines how far back she must lean? Draw a free-body diagram for the water skier to justify your answers.

Problem 11E A diving board 3.00 m long is supported at a point 1.00 m from the end, and a diver weighing 500 N stands at the free end (?Fig. E11.11?). The diving board is of uniform cross section and weighs 280 N. Find (a) the force at the support point and (b) the force at the left-hand end.

Problem 99CP Minimizing the tension. A heavy horizontal girder of length ?L has several objects suspended from it. It is supported by a frictionless pivot at its left end and a cable of negligible weight that is attached to an I-beam at a point a distance ?h directly above the girder’s center. Where should the other end of the cable be attached to the girder so that the cable’s tension is a minimum? (?Hint?: In evaluating and presenting your answer, don’t forget that the maximum distance of the point of attachment from the pivot is the length ?L? of the beam.)

Problem 12DQ In pioneer days, when a Conestoga wagon was stuck in the mud, people would grasp the wheel spokes and try to turn the wheels, rather than simply pushing the wagon. Why?

Problem 13DQ The mighty Zimbo claims to have leg muscles so strong that he can stand flat on his feet and lean forward to pick up an apple on the floor with his teeth. Should you pay to see him perform, or do you have any suspicions about his claim? Why?

Problem 14DQ Why is it easier to hold a 10-kg dumbbell in your hand at your side than it is to hold it with your arm extended horizontally?

Problem 12E A uniform aluminum beam 9.00 m long, weighing 300 N, rests symmetrically on two supports 5.00 m apart (?Fig. E11.12?). A boy weighing 600 N starts at point A and walks toward the right. (a) In the same diagram construct two graphs showing the upward forces FA and FB exerted on the beam at points A and B, as functions of the coordinate x of the boy. Let 1 cm = 100 N vertically, and 1 cm = 1.00 m horizontally. (b) From your diagram, how far beyond point B can the boy walk before the beam tips? (c) How far from the right end of the beam should support B be placed so that the boy can walk just to the end of the beam without causing it to tip?

Find the tension \(T\) in each cable and the magnitude and direction of the force exerted on the strut by the pivot in each of the arrangements in Fig. E11.13. In each case let \(w\) be the weight of the suspended crate full of priceless art objects. The strut is uniform and also has weight \(w\). Start each case with a free-body diagram of the strut.

Problem 14E The horizontal beam in ?Fig. E11.14 weighs 190 N, and its center of gravity is at its center. Find (a) the tension in the cable and (b) the horizontal and vertical components of the force exerted on the beam at the wall.

Problem 15DQ Certain features of a person, such as height and mass, are fixed (at least over relatively long periods of time). Are the following features also fixed? (a) location of the center of gravity of the body; (b) moment of inertia of the body about an axis through the person’s center of mass. Explain your reasoning.

Problem 16DQ During pregnancy, women often develop back pains from leaning backward while walking. Why do they have to walk this way?

Problem 15E Push-ups. To strengthen his arm and chest muscles, an 82-kg athlete who is 2.0 m tall is doing push-ups as shown in below Fig. His center of mass is 1.15 m from the bottom of his feet, and the centers of his palms are 30.0 cm from the top of his head. Find the force that the floor exerts on each of his feet and on each hand, assuming that both feet exert the same force and both palms do likewise. Begin with a free-body diagram of the athlete. Figure:

Suppose that you can lift no more than 650 N (around 150 lb) unaided. (a) How much can you lift using a 1.4-m-long wheelbarrow that weighs 80.0 N and whose center of gravity is 0.50 m from the center of the wheel (Fig. E11.16?)? The center of gravity of the load carried in the wheelbarrow is also 0.50 m from the center of the wheel. (b) Where does the force come from to enable you to lift more than 650 N using the wheelbarrow?

Problem 17DQ Why is a tapered water glass with a narrow base easier to tip over than a glass with straight sides? Does it matter whether the glass is full or empty?

Problem 17E You take your dog Clea to the vet, and the doctor decides he must locate the little beast’s center of gravity. It would be awkward to hang the pooch from the ceiling, so the vet must devise another method. He places Clea’s from feet on one scale and her hind feet on another. The front scale reads 157 N, while the rear scale reads 89 N. The vet next measures Clea and finds that her rear feet are 0.95 m behind her front feet. How much does Clea weigh and where is her center of gravity?

Problem 18DQ When a tall, heavy refrigerator is pushed across a rough floor, what factors determine whether it slides or tips?

Problem 18E A 15,000-N crane pivots around a friction-free axle at its base and is supported by a cable making a 25° angle with the crane (?Fig. E11.18?). The crane is 16 m long and is not uniform, its center of gravity being 7.0 m from the axle as measured along the crane. The cable is attached 3.0 m from the upper end of the crane. When the crane is raised to 55° above the horizontal holding an 11,000-N pallet of bricks by a 2.2-m, very light cord, find (a) the tension in the cable and (b) the horizontal and vertical components of the force that the axle exerts on the crane. Start with a free-body diagram of the crane.

Problem 19DQ If a metal wire has its length doubled and its diameter tripled, by what factor does its Young’s modulus change?

Problem 19E A 3.00-m-long, 240-N uniform rod at the zoo is held in the horizontal position by two ropes at its ends (Fig.). The left rope makes an angle of 150° with the rod and the right lope makes an angle ? with the horizontal. A 90-N howler monkey (Alouatta seniclulus) hangs motionless 0.50 m from the right end of the rod as he carefully studies you. Calculate the tensions in the two ropes and the angle ?. First make a free-body diagram of the rod. Figure:

Problem 20E A nonuniform beam 4.50 m long and weighing 1.00 kN makes an angle of 25.0° below the horizontal. It is held in position by a frictionless pivot at its upper right end and by a cable 3.00 m farther down the beam and perpendicular to it (Fig.). The center of gravity of the beam is 2.00 m down the beam from Ihe pivot. Lighting equipment exerts a 5.00-kN downward force 011 the lower left end of the beam. Find the tension ?T in the cable and the horizontal and vertical components of the force exerted on the beam by the pivot. Start by sketching a free-body diagram of the beam. Figure:

Problem 21DQ A metal wire of diameter D stretches by 0.100 mm when supporting a weight W. If the same-length wire is used to support a weight three times as heavy, what would its diameter have to be (in terms of D) so it still stretches only 0.100 mm?

Problem 22DQ Compare the mechanical properties of a steel cable, made by twisting many thin wires together, with the properties of a solid steel rod of the same diameter. What advantages does each have?

Problem 21E A Couple. Two forces equal in magnitude and opposite in direction, acting on an object at two different points, form what is called a couple. Two antiparallel forces with equal magnitudes F1 = F2 = 8.00 N are applied to a rod as shown in ?Fig. E11.21?. (a) What should the distance ?l between the forces be if they are to provide a net torque of 6.40 N ? m about the left end of the rod? (b) Is the sense of this torque clockwise or counterclockwise? (c) Repeat parts (a) and (b) for a pivot at the point on the rod where is applied.

Problem 22E BIO A Good Work out. You are doing exercises on a Nautilus machine in a gym to strengthen your deltoid (shoulder) muscles. Your arms are raised vertically and can pivot around the shoulder joint, and you grasp the cable of the machine in your hand 64.0 cm from your shoulder joint. The deltoid muscle is attached to the humerus 15.0 cm from the shoulder joint and makes a 12.0° angle with that bone (?Fig. E11.22?). If you have set the tension in the cable of the machine to 36.0 N on each arm, what is the tension in each deltoid muscle if you simply hold your outstretched arms in place? (Hint: Start by making a clear free-body diagram of your arm.)

Problem 23DQ The material in human bones and elephant bones is essentially the same, but an elephant has much thicker legs. Explain why, in terms of breaking stress.

Problem 23E BIO Neck Muscles. A student bends her head at 40.0° from the vertical while intently reading her physics book, pivoting the head around the upper vertebra (point P in ?Fig. E11.23?). Her head has a mass of 4.50 kg (which is typical), and its center of mass is 11.0 cm from the pivot point P. Her neck muscles are 1.50 cm from point P, as measured ?perpendicular to these muscles. The neck itself and the vertebrae are held vertical. (a) Draw a free-body diagram of the student’s head. (b) Find the tension in her neck muscles.

Problem 24DQ There is a small but appreciable amount of elastic hysteresis in the large tendon at the back of a horse’s leg. Explain how this can cause damage to the tendon if a horse runs too hard for too long a time.

Problem 24E BIO Biceps Muscle. A relaxed biceps muscle requires a force of 25.0 N for an elongation of 3.0 cm; the same muscle under maximum tension requires a force of 500 N for the same elongation. Find Young’s modulus for the muscle tissue under each of these conditions if the muscle is assumed to be a uniform cylinder with length 0.200 m and cross-sectional area 50.0 cm2.

When rubber mounting blocks are used to absorb machine vibrations through elastic hysteresis, as mentioned in Section 11.5, what becomes of the energy associated with the vibrations?

Problem 25E A circular steel wire 2.00 m long must stretch no more than 0.25 cm when a tensile force of 400 N is applied to each end of the wire. What minimum diameter is required for the wire?

Two circular rods, one steel and the other copper, are joined end to end. Each rod is 0.750 m long and 1.50 cm in diameter. The combination is subjected to a tensile force with magnitude 4000 N. For each rod, what are (a) the strain and (b) the elongation?

Problem 27E A metal rod that is 4.00 m long and 0.50 cm2 in cross-sectional area is found to stretch 0.20 cm under a tension of 5000 N. What is Young’s modulus for this metal?

Problem 28E Stress on a Mountaineer’s Rope. A nylon rope used by mountaineers elongates 1.10 m under the weight of a 65.0-kg climber. If the rope is 45.0 m in length and 7.0 mm in diameter, what is Young’s modulus for nylon?

Problem 29E P In constructing a large mobile, an artist hangs an aluminum sphere of mass 6.0 kg from a vertical steel wire 0.50 m long and 2.5 X 10-3 cm2 in cross-sectional area. On the bottom of the sphere he attaches a similar steel wire, from which he hangs a brass cube of mass 10.0 kg. For each wire, compute (a) the tensile strain and (b) the elongation.

Problem 31E BIO Compression of Human Bone. The bulk modulus for bone is 15 GPa. (a) If a diver-in-training is put into a pressurized suit, by how much would the pressure have to be raised (in atmospheres) above atmospheric pressure to compress her bones by 0.10% of their original volume? (b) Given that the pressure in the ocean increases by 1.0 X 104 Pa for every meter of depth below the surface, how deep would this diver have to go for her bones to compress by 0.10%? Does it seem that bone compression is a problem she needs to be concerned with when diving?

Problem 32E A solid gold bar is pulled up from the hold of the sunken RMS ?Titanic. (a) What happens to its volume as it goes from the pressure at the ship to the lower pressure at the ocean’s surface? (b) The pressure difference is proportional to the depth. How many times greater would the volume change have been had the ship been twice as deep? (c) The bulk modulus of lead is one-fourth that of gold. Find the ratio of the volume change of a solid lead bar to that of a gold bar of equal volume for the same pressure change.

Problem 33E Downhill Hiking. During vigorous downhill hiking, the force on the knee cartilage (the medial and lateral meniscus) can be up to eight times body weight. Depending on the angle of descent, this force can cause a large shear force on the cartilage and deform it. The cartilage has an area of about 10 cm2 and a shear modulus of 12 MPa. lf the hiker plus his pack have a combined mass of 110 kg (not unreasonable), and if the maximum force at impact is 8 times his body weight (which, of course, includes the weight of his pack) at an angle of 12° with the cartilage (below Fig.), through what angle (in degrees) will his knee cartilage be deformed? (Recall that the bone below the cartilage pushes upward with the same force as the downward force.) Figure:

Problem 34E A diving board 3.00 m long is supported at a point 1.00 m from the end, and a diver weighing 500 N stands at the free end (Fig. E11.11). The diving board is of uniform cross section and weighs 280 N. Find (a) the force at the support point and (b) the force at the left-hand end.

Problem 35E A specimen of oil having an initial volume of 600 cm3 is subjected to a pressure increase of 3.6 X 106 Pa, and the volume is found to decrease by 0.45 cm3. What is the bulk modulus of the material? The compressibility?

Problem 36E A square steel plate is 10.0 cm on a side and 0.500 cm thick. (a) Find the shear strain that results if a force of magnitude 9.0 X 105 N is applied to each of the four sides, parallel to the side. (b) Find the displacement x in centimeters.

Problem 37E A copper cube measures 6.00 cm on each side. The bottom face is held in place by very strong glue to a flat horizontal surface, while a horizontal force ?F ?is applied to the upper face parallel to one of the edges. (Consult Table 1.1.) (a) Show that the glue exerts a force ?F ?on the bottom face that is equal but opposite to the force on the top face. (b) How large must ?F ?be to cause the cube to deform by 0.250 mm? (c) If the same experiment were performed on a lead cube of the same size as the copper one, by what distance would it deform for the same force as in part (b)?

Problem 38E In lab tests on a 9.25-cm cube of a certain material, a force of 1375 N directed at 8.50° to the cube (?Fig. E11.37?) causes the cube to deform through an angle of 1.24°. What is the shear modulus of the material?

Problem 39E In a materials testing laboratory, a metal wire made from a new alloy is found to break when a tensile force of 90.8 N is applied perpendicular to each end. If the diameter of the wire is 1.84 mm, what is the breaking stress of the alloy?

Problem 40E A 4.0-m-long steel wire has a cross-sectional area of 0.050 cm2 .Its proportional limit has a value of 0.0016 times its Young’s modulus (see Table 11.1). Its breaking stress has a value of 0.0065 times its Young’s modulus. The wire is fastened at its upper end and hangs vertically. (a) How great a weight can be hung from the wire without exceeding the proportional limit?

Problem 41E CP A steel cable with cross-sectional area 3.00 cm2 has an elastic limit of 2.40 X 108 Pa. Find the maximum upward acceleration that can be given a 1200-kg elevator supported by the cable if the stress is not to exceed one-third of the elastic limit.

Problem 42E A brass wire is to withstand a tensile force of 350 N without breaking. What minimum diameter must the wire have?

A box of negligible mass rests at the left end of a 2.00-m, 25.0-kg plank (Fig. P11.43?). The width of the box is 75.0 cm, and sand is to be distributed uniformly throughout it. The center of gravity of the nonuniform plank is 50.0 cm from the right end. What mass of sand should be put into the box so that the plank balances horizontally on a fulcrum placed just below its midpoint?

Problem 44P A door 1.00 m wide and 2.00 m high weighs 280 N and is supported by two hinges, one 0.50 m from the top and the other 0.50 m from the bottom. Each hinge supports half the total weight of the door. Assuming that the door’s center or gravity is at its center, find the horizontal components of force exerted on the door by each hinge.

Problem 45P Mountain Climbing. Mountaineers often use a rope to lower them-selves down the face of a cliff (this is called ?rappelling?). They do this with their body nearly horizontal and their feet pushing against the cliff (?Fig. P11.45?). Suppose that an 82.0-kg climber, who is 1.90 m tall and has a center of gravity 1.1 m from his feet, rappels down a vertical cliff with his body raised 35.0° above the horizontal. He holds the rope 1.40 m from his feet, and it makes a 25.0° angle with the cliff face. (a) What tension does his rope need to support? (b) Find the horizontal and vertical components of the force that the cliff face exerts on the climber’s feet. (c) What minimum coefficient of static friction is needed to pre-vent the climber’s feet from slipping on the cliff face if he has one foot at a time against the cliff?

Problem 46P Sir Lancelot rides slowly out of the castle at Camelot and onto the 12.0-m-long drawbridge that passes over the moat (?Fig. P11.44?). Unbeknownst to him, his enemies have partially severed the vertical cable holding up the front end of the bridge so that it will break under a tension of 5.80 X 103 N. The bridge has mass 200 kg and its center of gravity is at its center. Lancelot, his lance, his armor, and his horse together have a combined mass of 600 kg. Will the cable break before Lancelot reaches the end of the drawbridge? If so, how far from the castle end of the bridge will the center of gravity of the horse plus rider be when the cable breaks?

Problem 47P Three vertical forces act on an airplane when it is flying at a constant altitude and with a constant velocity. These are the weight of the airplane. an aerodynamic force on the wing of the airplane, and an aerodynamic force on the airplane’s horizontal tail. (The aerodynamic forces are exerted by the surrounding air and are reactions to the forces that the wing and tail exert on the air as the airplane flies through it.) For a particular light airplane with a weight of 6700 N, the center of gravity is 0.30 m in front of the point where the wing’s vertical aerodynamic force acts and 3.66 m in front of the point where the tail’s vertical aerodynamic force acts. Determine the magnitude and direction (upward or downward) of each of the two vertical aero dynamic forces.

Problem 48P A pickup truck has a wheelbase of 3.00 m. Ordinarily, 10,780 N rests on the front wheels and 8820 N on the rear wheels when the truck is parked on a level road. (a) A box weighing 3600 N is now placed on the tailgate, 1.00 m behind the rear axle. How much total weight now rests on the front wheels? On the rear wheels? (b) How much weight would need to be placed on the tailgate to make the front wheels come off the ground?

Problem 49P A uniform, 255-N rod that is 2.00 m long carries a 225-N weight at its right end and an unknown weight W toward the left end (?Fig. P11.47?). When W is placed 50.0 cm from the left end of the rod, the system just balances horizontally when the fulcrum is located 75.0 cm from the right end. (a) Find W. (b) If W is now moved 25.0 cm to the right, how far and in what direction must the fulcrum be moved to restore balance?

Problem 50P A uniform, 8.0-m, 1150-kg beam is hinged to a wall and supported by a thin cable attached 2.0 m from the free end of the beam (?Fig. P11.46?). The beam is supported at an angle of 30.0° above the horizontal. (a) Draw a free-body diagram of the beam. (b) Find the tension in the cable. (c) How hard does the beam push inward on the wall?

Problem 51P You open a restaurant and hope to entice customers by hanging out a sign (below Fig.). The uniform horizontal beam supporting the sign is 1.50 m long, has a mass of 12.0 kg, and is hinged to the wall. The sign itself is uniform with a mass of 28.0 kg and overall length of 1.20 m. The two wires supporting the sign are each 32.0 cm long, are 90.0 cm apart, and are equally spaced from the middle of the sign. The cable supporting the beam is 2.00 m long. (a) What minimum tension must your cable be able to support without having your sign come crashing clown? (b) What minimum vertical force must the hinge be able to support without pulling out of the wall? Figure:

A claw hammer is used to pull a nail out of a board (Fig. P11.48). The nail is at an angle of \(60^{\circ}\) to the board, and a force \(\overrightarrow{\boldsymbol{F}}_{1}\) of magnitude 400 N applied to the nail is required to pull it from the board. The hammer head contacts the board at point A, which is 0.080 m from where the nail enters the board. A horizontal force \(\overrightarrow{\boldsymbol{F}}_{2}\) is applied to the hammer handle at a distance of 0.300 m above the board. What magnitude of force \(\overrightarrow{\boldsymbol{F}}_{2}\) is required to apply the required 400-N force \(\left(F_{1}\right)\) to the nail? (Ignore the weight of the hammer.)

Problem 53P End A of the bar AB in ?Fig. P11.50 rests on a friction-less horizontal surface, and end B is hinged. A horizontal force of magnitude 220 N is exerted on end A. Ignore the weight of the bar. What are the horizontal and vertical components of the force exerted by the bar on the hinge at B?

Problem 54P A museum of modern art is displaying an irregular 426-N sculpture by hanging it from two thin vertical wires, ?A and ?B?, that are 1.25 m apart (Fig.). The center of gravity of this piece of art is located 48.0 cm from its extreme right tip. Find the tension in each wire. Figure:

Problem 57P BIO Leg Raises. In a simplified version of the musculature action in leg raises, the abdominal muscles pull on the femur (thigh bone) to raise the leg by pivoting it about one end (?Fig. P11.53?). When you are lying horizontally, these muscles make an angle of approximately 5° with the femur, and if you raise your legs, the muscles remain approximately horizontal, so the angle ? increases. Assume for simplicity that these muscles attach to the femur in only one place, 10 cm from the hip joint (although, in reality, the situation is more complicated). For a certain 80-kg person having a leg 90 cm long, the mass of the leg is 15 kg and its center of mass is 44 cm from his hip joint as measured along the leg. If the per-son raises his leg to 60° above the horizontal, the angle between the abdominal muscles and his femur would also be about 60°. (a) With his leg raised to 60°, find the tension in the abdominal muscle on each leg. Draw a free-body diagram. (b) When is the tension in this muscle greater: when the leg is raised to 60° or when the person just starts to raise it off the ground? Why? (Try this yourself.) (c) If the abdominal muscles attached to the femur were perfectly horizontal when a person was lying down, could the person raise his leg? Why or why not?

Problem 56P A Truck on a Drawbridge. A loaded cement mixer drives onto an old drawbridge, where it stalls with its center of gravity three-quarters of the way across the span. The truck driver radios for help, sets the handbrake, and waits. Meanwhile, a boat approaches, so the drawbridge is raised by means of a cable attached to the end opposite the hinge (?Fig. P11.52?). The drawbridge is 40.0 m long and has a mass of 18,000 kg; its center of gravity is at its midpoint. The cement mixer, with driver, has mass 30,000 kg. When the drawbridge has been raised to an angle of 30o above the horizontal, the cable makes an angle of 70o with the surface of the bridge. (a) What is the tension T in the cable when the drawbridge is held in this position? (b) What are the horizontal and vertical components of the force the hinge exerts on the span?

Problem 58P A non uniform fire escape ladder is 6.0 m long when extended to the icy alley below. It is held at the top by a frictionless pivot, and there is negligible frictional force from the icy surface at the bottom. The ladder weighs 250 N, and its center of gravity is 2.0 m along the ladder from its bottom. A mother and child of total weight 750 N are on the ladder 1.5 m Crom the pivot. The ladder makes an angle 0 with the horizontal. Find the magnitude and direction of (a) the force exerted by the icy alley on the ladder and (b) the force exerted by the ladder on the pivot. (c) Do your answers in parts (a) and (b) depend on the angle ????

Problem 60P You are asked to design the decorative mobile shown in ?Fig. P11.56?. The strings and rods have negligible weight, and the rods are to hang horizontally. (a) Draw a free-body diagram for each rod. (b) Find the weights of the balls A, B, and C. Find the tensions in the strings S1, S2, and S3. (c) What can you say about the horizontal location of the mobile’s center of gravity? Explain.

Problem 55P BIO Supporting a Broken Leg. A therapist tells a 74-kg patient with a broken leg that he must have his leg in a cast suspended horizontally. For minimum discomfort, the leg should be supported by a vertical strap attached at the center of mass of the leg–cast system (?Fig. P11.51?). To comply with these instructions, the patient consults a table of typical mass distributions and finds that both upper legs (thighs) together typically account for 21.5% of body weight and the center of mass of each thigh is 18.0 cm from the hip joint. The patient also reads that the two lower legs (including the feet) are 14.0% of body weight, with a center of mass 69.0 cm from the hip joint. The cast has a mass of 5.50 kg, and its center of mass is 78.0 cm from the hip joint. How far from the hip joint should the supporting strap be attached to the cast?

Problem 59P A uniform strut of mass ?m makes an angle ?? with the horizontal. It is supported by a frictionless pivot located at one third its length from its lower left end and a horizontal rope at its upper tight end. A cable and package of total weight ?w hang from its upper right end. (a) Find the vertical and horizontal components ?V and ?H of the pivot’s force on the strut as well as the tension ?T in the rope. (b) If the maximum safe tension in the rope is 700 N and the mass of the strut is 30.0 kg, find the maximum safe weight of the cable and package when the strut makes an angle of 55.0° with the horizontal. (c) For what angle ?? can no weight be safely suspended from the right end of the strut?

Problem 62P CP A uniform drawbridge must be held at a 37o angle above the horizontal to allow ships to pass underneath. The drawbridge weighs 45,000 N and is 14.0 m long. A cable is connected 3.5 m from the hinge where the bridge pivots (measured along the bridge) and pulls horizontally on the bridge to hold it in place. (a) What is the tension in the cable? (b) Find the magnitude and direction of the force the hinge exerts on the bridge. (c) If the cable suddenly breaks, what is the magnitude of the angular acceleration of the drawbridge just after the cable breaks? (d) What is the angular speed of the drawbridge as it becomes horizontal?

Problem 61P A uniform, 7.5-m-long beam weighing 5860 N is hinged to a wall and supported by a thin cable attached 1.5 m from the free end of the beam. The cable runs between the beam and the wall and makes a 40° angle with the beam. What is the tension in the cable when the beam is at an angle of 30° above the horizontal?

Problem 63P BIO Tendon-Stretching Exercises. As part of an exercise program, a 75-kg person does toe raises in which he raises his entire body weight on the ball of one foot (?Fig. P11.59?). The Achilles tendon pulls straight upward on the heel bone of his foot. This tendon is 25 cm long and has a cross-sectional area of 78 mm2 and a Young’s modulus of 1470 MPa. (a) Draw a free-body diagram of the person’s foot (everything below the ankle joint). Ignore the weight of the foot. (b) What force does the Achilles tendon exert on the heel during this exercise? Express your answer in newtons and in multiples of his weight. (c) By how many millimeters does the exercise stretch his Achilles tendon?

Problem 64P (a) In ?Fig. P11.60 a 6.00-m-long, uniform beam is hanging from a point 1.00 m to the right of its center. The beam weighs 140 N and makes an angle of 30.0o with the vertical. At the right-hand end of the beam a 100.0-N weight is hung; an un-known weight w hangs at the left end. If the system is in equilibrium, what is w? You can ignore the thickness of the beam. (b) If the beam makes, instead, an angle of 45.0o with the vertical, what is w?

Problem 65P A uniform, horizontal flagpole 5.00 m long with a weight of 200 N is hinged to a vertical wall at one end. A 600-N stuntwoman hangs from its other end. The flagpole is supported by a guy wire running from its outer end to a point on the wall directly above the pole. (a) If the tension in this wire is not to exceed 1000 N, what is the minimum height above the pole at which it may be fastened to the wall? (b) If the flagpole remains horizontal, by how many newtons would the tension be increased if the wire were fastened 0.50 m below this point?

Problem 67P BIO Downward-Facing Dog. The yoga exercise “Downward-Facing Dog” requires stretching your hands straight out above your head and bending down to lean against the floor. This exercise is performed by a 750-N person as shown in Fig. P11.63?. When he bends his body at the hip to a 90° angle between his legs and trunk, his legs, trunk, head, and arms have the dimensions indicated. Furthermore, his legs and feet weigh a total of 277 N, and their center of mass is 41 cm from his hip, measured along his legs. The person’s trunk, head, and arms weigh 473 N, and their center of gravity is 65 cm from his hip, measured along the upper body. (a) Find the normal force that the floor exerts on each foot and on each hand, assuming that the person does not favor either hand or either foot. (b) Find the friction force on each foot and on each hand, assuming that it is the same on both feet and on both hands (but not necessarily the same on the feet as on the hands). [?Hint: First treat his entire body as a system; then isolate his legs (or his upper body).]

Problem 68P When you stretch a wire, rope, or rubber band, it gets thinner as well as longer. When Hooke’s law holds, the fractional decrease in width is proportional to the tensile strain. If must be applied perpendicular to each end of the cylinder to cause its radius to decrease by 0.10 mm? Assume that the breaking stress and proportional limit for the metal are extremely large and are not exceeded.

Problem 66P A holiday decoration consists of two shiny glass spheres with masses 0.0240 kg and 0.0360 kg suspended from a uniform rod with mass 0.120 kg and length 1.00 m (?Fig. P11.62?). The rod is suspended from the ceiling by a vertical cord at each end, so that it is horizontal. Calculate the tension in each of the cords A through F.

Problem 69P A worker wants to turn over a uniform, 1250-N, rectangular crate by pulling at 53.0° on one of its vertical sides (?Fig. P11.65?). The floor is rough enough to pre-vent the crate from slipping. (a) What pull is needed to just start the crate to tip? (b) How hard does the floor push upward on the crate? (c) Find the friction force on the crate. (d) What is the minimum coefficient of static friction needed to prevent the crate from slipping on the floor?

Problem 70P One end of a uniform meter stick is placed against a vertical wall (?Fig. P11.66?). The other end is held by a lightweight cord that makes an angle ? with the stick. The coefficient of static friction between the end of the meter stick and the wall is 0.40. (a) What is the maximum value the angle ? can have if the stick is to remain in equilibrium? (b) Let the angle ? be 15o. A block of the same weight as the meter stick is suspended from the stick, as shown, at a distance x from the wall. What is the minimum value of x for which the stick will remain in equilibrium? (c) When ? = 15o, how large must the coefficient of static friction be so that the block can be attached 10 cm from the left end of the stick without causing it to slip?

Problem 71P Two friends are carrying a 200-kg crate up a flight of stairs. The crate is 1.25 m long and 0.500 m high, and its center of gravity is at its center. The stairs make a 45.0o angle with respect to the floor. The crate also is carried at a 45.0o angle, so that its bottom side is parallel to the slope of the stairs (?Fig. P11.67?). If the force each person applies is vertical, what is the magnitude of each of these forces? Is it better to be the person above or below on the stairs?

Problem 72P BIO Forearm. I ? n the human arm, the forearm and hand pivot about the elbow joint. Consider a simplified model in which the biceps muscle is attached to the forearm 3.80 cm from the elbow joint. Assume that the person’s hand and forearm together weigh 15.0 N and that their center of gravity is 15.0 cm from the elbow (not quite halfway to the hand). The forearm is held horizontally at a right angle to the upper arm, with the biceps muscle exerting its force perpendicular to the forearm. (a) Draw a free-body diagram for the forearm, and find the force exerted by the biceps when the hand is empty. (b) Now the person holds an 80.0-N weight in his hand, with the forearm still horizontal. Assume that the center of gravity of this weight is 33.0 cm from the elbow. Draw a free-body diagram for the forearm, and find the force now exerted by the biceps. Explain why the biceps muscle needs to be very strong. (c) Under the conditions of part (b), find the magnitude and direction of the force that the elbow joint exerts on the forearm. (d) While holding the 80.0-N weight, the person raises his forearm until it is at an angle of 53.0o above the horizontal. If the biceps muscle continues to exert its force perpendicular to the forearm, what is this force now? Has the force increased or decreased from its value in part (b)? Explain why this is so, and test your answer by doing this with your own arm.

Problem 73P BIO CALC R ? efer to the discussion of holding a dumbbell in Example 11.4 (Section 11.3). The maximum weight that can be held in this way is limited by the maximum allowable tendon tension ?T ?(determined by the strength of the tendons) and by the distance ?D ?from the elbow to where the tendon attaches to the forearm. (a) Let represent the maximum value of the tendon tension. Use the results of Example 11.4 to express is positive or negative. (c) A chimpanzee tendon is attached to the forearm at a point farther from the elbow than for humans. Use this to explain why chimpanzees have stronger arms than humans. (The disadvantage is that chimpanzees have less flexible arms than do humans.)

Problem 75P Flying Buttress. (a) A symmetric building has a roof sloping upward at 35.0° above the horizontal on each side. If each side of the uniform roof weighs 10,000 N. find the horizontal force that this roof exerts at the top of the wall, which tends to push out the walls. Which type of building would be more in clanger of collapsing: one with tall walls or one with short walls? Explain. (b) As you saw in part (a), tall walls are in danger of collapsing from the weight of the roof. This problem plagued the ancient builders of large structures. A solution used in the great Gothic cathedrals during the 1200s was the flying buttress, a stone support running between the walls and the ground that helped to hold in the walls. A Gothic church has a uniform roof weighing a total of 20,000 N and rising at 40° above the horizontal at each wall. The walls are 40 m tall, and a flying buttress meets each wall 10 m below the base of the roof. What horizontal force must this flying buttress apply to the wall?

Problem 74P A uniform, 90.0-N table is 3.6 m long, 1.0 m high, and 1.2 m wide. A 1500-N weight is placed 0.50 m from one end of the table, a distance of 0.60 m from each side of the table. Draw a freebody diagram for the table and find the force that each of the four legs exerts on the floor.

You are trying to raise a bicycle wheel of mass m and radius R up over a curb of height h. To do this, you apply a horizontal force \(\overrightarrow{\boldsymbol{F}}\) (Fig. P11.72). What is the smallest magnitude of the force \(\overrightarrow{\boldsymbol{F}}\) that will succeed in raising the wheel onto the curb when the force is applied (a) at the center of the wheel and (b) at the top of the wheel? (c) In which case is less force required?

Problem 77P The Farmyard Gate. A gate 4.00 m wide and 2.00 m high weighs 500 N. Its center of gravity is at its center, and it is hinged at ?A and ?B?. To relieve the strain on the top hinge, a wire ?CD is connected as shown in below Fig. The tension in ?CD is increased until the horizontal force at hinge ?A is zero. (a) What is the tension in the wire ?CD?? (b) What is the magnitude of the horizontal component of the force at hinge ?B?? (c) What is the combined vertical force exerted by hinges ?A? and ?B?? Figure:

Problem 79P Two uniform, 75.0-g marbles 2.00 cm in diameter are stacked as shown in ?Fig. P11.75 in a container that is 3.00 cm wide. (a) Find the force that the container exerts on the marbles at the points of contact A, B, and C. (b) What force does each marble exert on the other?

Problem 80P Two identical, uniform beams weighing 260 N each are connected at one end by a frictionless hinge. A light horizontal crossbar attached at the midpoints of the beams maintains an angle of 53.0° between the beams. The beams are suspended from the ceiling by vertical wires such that they form a “V” (?Fig. P11.76?). (a) What force does the crossbar exert on each beam? (b) Is the crossbar under tension or compression? (c) What force (magnitude and direction) does the hinge at point A exert on each beam?

Problem 81P An engineer is de-signing a conveyor system for loading hay bales into a wagon (?Fig. P11.77?). Each bale is 0.25 m wide, 0.50 m high, and 0.80 m long (the dimension perpendicular to the plane of the figure), with mass 30.0 kg. The center of gravity of each bale is at its geometrical center. The coefficient of static friction between a bale and the conveyor belt is 0.60, and the belt moves with constant speed. (a) The angle ? of the conveyor is slowly increased. At some critical angle a bale will tip (if it doesn’t slip first), and at some different critical angle it will slip (if it doesn’t tip first). Find the two critical angles and determine which happens at the smaller angle. (b) Would the out-come of part (a) be different if the coefficient of friction were 0.40?

A weight W is supported by attaching it to a vertical uniform metal pole by a thin cord passing over a pulley having negligible mass and friction. The cord is attached to the pole 40.0 cm below the top and pulls horizontally on it (Fig. P11.78). The pole is pivoted about a hinge at its base, is 1.75 m tall, and weighs 55.0 N. A thin wire connects the top of the pole to a vertical wall. The nail that holds this wire to the wall will pull out if an outward force greater than 22.0 N acts on it. (a) What is the greatest weight W that can be supported this way without pulling out the nail? (b) What is the magnitude of the force that the hinge exerts on the pole?

Problem 83P A garage door is mounted on an overhead rail (?Fig. P11.79?). The wheels at A and B have rusted so that they do not roll, but rather slide along the track. The coefficient of kinetic friction is 0.52. The distance between the wheels is 2.00 m, and each is 0.50 m from the vertical sides of the door. The door is uniform and weighs 950 N. It is pushed to the left at constant speed by a horizontal force (a) If the distance ?h is 1.60 m, what is the vertical component of the force exerted on each wheel by the track? (b) Find the maximum value h can have without causing one wheel to leave the track.

Problem 84P A horizontal boom is supported at its left end by a frictionless pivot. It is held in place by a cable attached to the right hand end of the boom. A chain and crate of total weight w? hang from somewhere along the boom. The boom’s weight ?w?b cannot be ignored and the boom may or may not be uniform. (a) Show that the tension in the cable is the same whether the cable makes an angle ?? or an angle 180° ? ?? with the horizontal, and that the horizontal force component exerted on the boom by the pivot has equal magnitude but opposite direction for the two angles. (b) Show that the cable cannot be horizontal. (c) Show that the tension in the cable is a minimum when the cable is vertical, pulling upward on the right end of the boom. (d) Show that when the cable is vertical, the force exerted by the pivot on the boom is vertical.

Problem 85P Prior to being placed in its hole, a 5700-N, 9.0-m-long, uniform utility pole makes some nonzero angle with the vertical. A vertical cable attached 2.0 m below its upper encl holds it in place while its lower end rests on the ground. (a) Find the tension in the cable and the magnitude and direction of the force exerted by the ground on the pole. (b) Why don’t we need to know the angle the pole makes with the vertical, as long as it is not zero?

Problem 86P Pyramid Builders. Ancient pyramid builders are balancing a uniform rectangular slab of stone tipped at an angle ? above the horizontal using a rope (?Fig. P11.80?). The rope is held by five workers who share the force equally. (a) If ? = 20.0o, what force does each worker exert on the rope? (b) As ? increases, does each worker have to exert more or less force than in part (a), assuming they do not change the angle of the rope? Why? (c) At what angle do the workers need to exert ?no force to balance the slab? What happens if ? exceeds this value?

Problem 87P You hang a floodlamp from the end of a vertical steel wire. The floodlamp stretches the wire 0.18 mm and the stress is proportional to the strain. How much would it have stretched (a) if the wire were twice as long? (b) if the wire had the same length but twice the diameter. (c) for a copper wire of the orIginal length and diameter?

Problem 88P Hooke’s Law for a Wire. A ? wire of length where Y? ?is Young’s modulus for the material of which the wire is made. (b) What would the force constant be for a 75.0-cm length of 16-gauge (diameter = 1.297 mm) copper wire? See Table 11.1. (c) What would ?W h ? ave to be to stretch the wire in part (b) by 1.25 mm?

Problem 89P A 12.0-kg mass. fastened to the end of an aluminium wire with an un stretched length of 0.50 m, is whirled in a vertical circle with a constant angular speed of 120 rev/min. The crosssectional area of the wire is 0.014 cm2. Calculate the elongation of the wire when the mass is (a) at the lowest point of the path and (b) at the highest point of its path.

Problem 90P A metal wire 3.50 m long and 0.70 mm in diameter was given the following test. A load weighing 20 N was originally hung from the wire to keep it taut. The position of the lower end of the wire was read on a scale as load was added. A dd Scale ed Readi Lo ng ad (cm) (N ) 0 3.02 10 3.07 20 3.12 30 3.17 40 3.22 50 3.27 60 3.32 70 4.27 (a) Graph these values, plotting the increase in length horizontally and the added load vertically. (b) Calculate the value of Young’s modulus. (c) The proportional limit occured at a scale reading of 3.34 cm. What was the stress at this point?

Problem 91P A 1.05-m-long rod of negligible weight is supported at its ends by wires A and B of equal length (?Fig. P11.83?). The cross-sectional area of A is 2.00 mm2 and that of B is 4.00 mm2. Young’s modulus for wire A is 1.80 X 1011 Pa; that for B is 1.20 X 1011 Pa. At what point along the rod should a weight w be suspended to produce (a) equal stresses in A and B and (b) equal strains in A and B?

Problem 92P An amusement park ride consists of airplane shaped cars attached to steel rods (below figure). Each rod has a length of 15.0 m and a cross sectional area of 8.00 cm2. (a) How much is the rod stretched when the ride is at rest? (Assume that each car plus two people seated in it has a total weight of 1900 N.) (b) When operating, the ride has a maximun angular speed of 8.0 rev/min. How much is the rod stretched then? Figure:

Problem 93P A brass rod with a length of 1.40 m and a cross-sectional area of 2.00 cm2 is fastened end to end to a nickel rod with length ?L and cross-sectional area 1.00 cm2. The compound rod is subjeced to equal and opposite pulls of magnitude 4.00 × 104 N at its ends. (a) Find the length L of the nickel rod if the elongations of the two rods are equal. (b) What is the stress in each rod? (c) What is the strain in each rod?

Problem 94P CP BIO Stress on the Shin Bone. The compressive strength of our bones is important in everyday life. Young’s modulus for bone is about 1.4 X 1010 Pa. Bone can take only about a 1.0% change in its length before fracturing. (a) What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 3.0 cm2? (This is approximately the cross-sectional area of a tibia, or shin bone, at its narrowest point.) (b) Estimate the maximum height from which a 70-kg man could jump and not fracture his tibia. Take the time between when he first touches the floor and when he has stopped to be 0.030 s, and assume that the stress on his two legs is distributed equally.

Problem 95P A moonshiner produces pure ethanol (ethyl alcohol) late at night and stores it in a stainless steel tank in the form of a cylinder 0.300 m in diameter with a tight-fitting piston at the top. The total volume of the tank is 250 L(0.250 m3). In an attempt to squeeze a little more into the tank, the moonshiner piles 1420 kg of lead bricks on top of the piston. What additional volume of ethanol can the moonshiner squeeze into the tank? (Assume that the wall of the tank is perfectly rigid.)

Problem 97CP A bookcase weighing 1500 N rests on a horizontal surface for which the coefficient of static fiction is ???s = 0.40. The bookcase is 1.80 m tall and 2.00 m wide; its center of gravity is at its geometrical center. The bookcase rests on four short legs that are each 0.10 m from the edge of the bookcase. A Person pulls on a rope attached to an upper corner of the bookcase. with a force that makes an angle ?? with the bookcase. (fig.). (a) If ?? = 90° so is horizontal. Show that as is increased from zero, the bookcase will start to slide before it tips, and calculate the magnitude of that will start the bookcase sliding. (b) If ?? = 0°, so is vertical, show that the bookcase will tip over rather than slide, and calculate the magnitude of that will cause the bookcase to start to tip. (c) Calculate as a function of ? the magnitude of that will cause the bookcase to start to slide and the magnitude that will cause it to start to tip. What is the smallest value that ?? can have so that the bookcase will still start to slide before it starts to tip? Figure:

Problem 98CP Knocking Over a Post. One end of a post weighing 400 N and with height h rests on a rough horizontal surface with µs = 0.30. The upper end is held by a rope fastened to the surface and making an angle of 36.9o with the post (?Fig. P11.90?). A horizontal force is exerted on the post as shown. (a) If the force is applied at the midpoint of the post, what is the largest value it can have without causing the post to slip? (b) How large can the force be without causing the post to slip if its point of application is of the way from the ground to the top of the post? (c) Show that if the point of application of the force is too high, the post cannot be made to slip, no matter how great the force. Find the critical height for the point of application.

Two ladders, 4.00 m and 3.00 m long, are hinged at point A and tied together by a horizontal rope 0.90 m above the floor (Fig. P11.96). The ladders weigh 480 N and 360 N, respectively, and the center of gravity of each is at its center. Assume that the floor is freshly waxed and frictionless. (a) Find the upward force at the bottom of each ladder. (b) Find the tension in the rope. (c) Find the magnitude of the force one ladder exerts on the other at point A. (d) If an 800-N painter stands at point A, find the tension in the horizontal rope.