## 11.47 .. 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?

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## 11.46 .. 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?

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## 11.45 … Mountain Climbing. Mountaineers often use a rope to lower themselves 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 prevent the climber’s feet from slipping on the cliff face if he has one foot at a time against the cliff?

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## 11.44 . 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 * 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?

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## 11.43 … 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?

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## 11.42 … A door 1.00 m wide and 2.00 m high weighs 330 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 of gravity is at its center find the horizontal components of force exerted on the door by each hinge.

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## 11.41 .. CP A steel cable with cross-sectional area 3.00 cm2 has an elastic limit of 2.40 * 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.

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## 11.40 . 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? (b) How much will the wire stretch under this load? (c) What is the maximum weight that the wire can support?

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## 11.39 .. 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?

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

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## 11.37 . 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?

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## 11.36 .. 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 * 105 N is applied to each of the four sides parallel to the side. (b) Find the displacement x in centimeters.

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## 11.35 .. 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 11.1.) (a) Show that the glue exerts a force F on the bottom face that is equal in magnitude 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)?

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## 11.34 .. In the Challenger Deep of the Marianas Trench the depth of seawater is 10.9 km and the pressure is 1.16 * 108 Pa (about 1.15 * 103 atm). (a) If a cubic meter of water is taken from the surface to this depth what is the change in its volume? (Normal atmospheric pressure is about 1.0 * 105 Pa. Assume that k for seawater is the same as the freshwater value given in Table 11.2.) (b) What is the density of seawater at this depth? (At the surface seawater has a density of 1.03 * 103 kg>m3.)

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

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## 11.32 . 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.

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## 11.31 .. 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 * 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?

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## 11.30 .. A vertical solid steel post 25 cm in diameter and 2.50 m long is required to support a load of 8000 kg. You can ignore the weight of the post. What are (a) the stress in the post; (b) the strain in the post; and (c) the change in the post’s length when the load is applied?

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## 11.29 .. 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 * 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.

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## 11.28 .. 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?

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## 11.27 .. A metal rod that is 4.00 m long and 0.50 cm2 in crosssectional area is found to stretch 0.20 cm under a tension of 5000 N. What is Young’s modulus for this metal?

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## 11.26 .. 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?

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## 11.25 .. A circular steel wire 2.00 m long must stretch no more than 0.25 cm when a tensile force of 700 N is applied to each end of the wire. What minimum diameter is required for the wire?

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## 11.24 . 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.

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## 11.23 .. 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.

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## 11.22 .. 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.)

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## 11.21 . 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 F S 2 is applied.

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