Answer: A 2500-kg trailer is attached to a stationary

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

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.

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?

Your doctors scale has arms on which weights slide to counter your weight, Fig. 12-39. These weights are much lighter than you are. How does this work?

A ground retaining wall is shown in Fig. 12-40a. The ground, particularly when wet, can exert a significant force F on the wall, (a) What force produces the torque to keep the wall upright? (b) Explain why the retaining wall in Fig. 12-40b would be much less likely to overturn than that in Fig. 12-40a.

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

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 25-cm mark is in equilibrium when a 1 -kg rock is suspended at the 0-cm end (as shown in Fig. 12-41). Is the mass of the meter stick greater than, equal to, or less than the mass of the rock? Explain your reasoning.

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

Figure 12-42 shows a cone. Explain how to lay it on a flat table so that it is in (a) stable equilibrium, (b) unstable equilibrium, (c) neutral equilibrium.

Place yourself facing the edge of an open door. Position your feet astride the door with your nose and abdomen touching the doors edge. Try to rise on your tiptoes. Why cant this be done?

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

Which of the configurations of brick, (a) or (b) of Fig. 12-43, is the more likely to be stable? Why?

Name the type of equilibrium for each position of the ball in Fig. 12-44.

Is the Youngs modulus for a bungee cord smaller or larger than that for an ordinary rope?

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

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. 12-7? If so, which one(s)? Explain.

(II) The Achilles tendon is attached to the rear of the foot as shown in Fig. 12-58. When a person elevates himself just barely off the floor on the ball of one foot, estimate the tension FT in the Achilles tendon (pulling upward), and the (downward) force Fb exerted by the lower leg bone on the foot. Assume the person has a tendon mass of 72 kg and D is twice as long as d.

(II) A shop sign weighing 215 N is supported by a uniform 155-N beam as shown in Fig. 12-59. 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. 12-60. The uniform aluminum pole AB is 7.20 m long and has a mass of 12.0 kg. The mass of the traffic light is 21.5 kg. Determine (a) the tension in the horizontal massless cable CD, and (b) the vertical and horizontal components of the force exerted by the pivot A on the aluminum pole.

(II) A uniform steel beam has a mass of 940 kg. On it is resting half of an identical beam, as shown in Fig. 12-61. What is the vertical support force at each end?

(II) Two wires run from the top of a pole 2.6 m tall that supports a volleyball net. The two wires are anchored to the ground 2.0 m apart, and each is 2.0 m from the pole (Fig. 12-62). The tension in each wire is 115 N. What is the tension in the net, assumed horizontal and attached at the top of the pole?

(II) A large 62.0-kg board is propped at a 45 angle against the edge of a barn door that is 2.6 m wide. How great a horizontal force must a person behind the door exert (at the edge) in order to open it? Assume that there is negligible friction between the door and the board but that the board is firmly set against the ground.

(II) Repeat Problem 24 assuming the coefficient of friction between the board and the door is 0.45.

(II) A 0.75-kg sheet hangs from a massless clothesline as shown in Fig. 12-63. The clothesline on either side of the sheet makes an angle of 3.5 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?

(II) A uniform rod AB of length 5.0 m and mass M = 3.8 kg is hinged at A and held in equilibrium by a light cord, as shown in Fig. 12-64. A load W = 22 N hangs from the rod at a distance x so that the tension in the cord is 85 N. (a) Draw a free-body diagram for the rod. (b) Determine the vertical and horizontal forces on the rod exerted by the hinge, (c) Determine d from the appropriate torque equation.

(Ill) A 56.0-kg person stands 2.0 m from the bottom of the stepladder shown in Fig. 12-65. Determine (a) the tension in the horizontal tie rod, which is halfway up the ladder, (b) the normal force the ground exerts on each side of the ladder, and (c) the force (magnitude and direction) that the left side of the ladder exerts on the right side at the hinge on the top. Ignore the mass of the ladder and assume the ground is frictionless. [Hint. Consider free-body diagrams for each section of the ladder.]

(Ill) A door 2.30 m high and 1.30 m wide has a mass of 13.0 kg. A hinge 0.40 m from the top and another hinge 0.40 m from the bottom each support half the doors weight (Fig. 12-66). 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.

(Ill) A cubic crate of side s = 2.0 m is top-heavy: its cg is 18 cm above its true center. How steep an incline can the crate rest on without tipping over?

(Ill) A refrigerator is approximately a uniform rectangular solid 1.9 m tall, 1.0 m wide, and 0.75 m deep. If it sits upright on a truck with its 1 .0-m dimension in the direction of travel, and if the refrigerator cannot slide on the truck, how rapidly can the truck accelerate without tipping the refrigerator over? [Hint. The normal force would act at one comer.]

(Ill) A uniform ladder of mass m and length leans at an angle 6 against a frictionless wall, Fig. 12-67. If the coefficient of static friction between the ladder and the ground is [jl s , determine a formula for the minimum angle at which the ladder will not slip.

(II) 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 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?

(I) A marble column of cross-sectional area 1.4 m2 supports a mass of 25,000 kg. (a) What is the stress within the column? (b) What is the strain?

(I) By how much is the column in Problem 35 shortened if it is 8.6 m high?

(I) A sign (mass 1700 kg) hangs from the end of a vertical steel girder with a cross-sectional area of 0.012 m2. (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.)

(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 (l.O X 105N/m2).

(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 8.5 mm. Calculate Youngs modulus of this tendon.

(II) At depths of 2000 m in the sea, the pressure is about 200 times atmospheric pressure (l atm = 1.0 X 105N/m2). By what percentage does the interior space of an iron bathyspheres volume change at this depth?

(Ill) A pole projects horizontally from the front wall of a shop. A 6.1-kg sign hangs from the pole at a point 2.2 m from the wall (Fig. 12-68). (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).

(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?

(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?

(II) If a compressive force of 3.3 X 104 N is exerted on the end of a 22-cm-long bone of cross-sectional area 3.6 cm2, (a) will the bone break, and (b) if not, by how much does it shorten?

(II) (a) What is the minimum cross-sectional area required of a vertical steel cable from which is suspended a 270-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. 12-69 (ra = 2900 kg) are made of wood. Calculate the minimum cross-sectional area required of each, assuming a safety factor of 9.0.

(II) An iron bolt is used to connect two iron plates together. The bolt must withstand shear forces up to about 3300 N. Calculate the minimum diameter for the bolt, based on a safety factor of 7.0.

(Ill) 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 8.0.

(II) A heavy load Mg = 66.0 kN hangs at point E of the single cantilever truss shown in Fig. 12-70. (a) Use a torque equation for the truss as a whole to determine the tension Ft in the support cable, and then determine the force FA on the truss at pin A. (b) Determine the force in each member of the truss. Neglect the The weight of the trusses, which is small compared to the load.

(II) Figure 12-71 shows a simple truss that carries a load at the center (C) of 1.35 X 104N. g (a) Calculate the force on each strut at the pins, A, B, C, D, and (b) determine which struts (ignore their masses) are under tension and which under compression.

(II) (a) What minimum cross-sectional area must the trusses have in Example 12-11 if they are of steel (and all the same size for looks), using a safety factor of 7.0? (b) If at any time the bridge may carry as many as 60 trucks with an average

(II) Consider again Example 12-11 but this time assume the roadway is supported uniformly so that \ its mass M (= 7.0 X 105 kg) acts at the center and \ M at each end support (think of the bridge as two spans, AC and CE, so the center pin supports two span ends). Calculate the magnitude of the force in each truss member and compare to Example 12-11.

(Ill) The truss shown in Fig. 12-72 supports a railway bridge. Determine the compressive or tension force in each strut if a 53-ton (l ton = 103 kg) train locomotive is stopped at the midpoint between the center and one end. Ignore the masses of the rails and truss, and use only \ the mass of train because there are two trusses (one on each side of the train). Assume all triangles are equilateral. [Hint: See Fig. 12-29.]

(Ill) Suppose in Example 12-11, a 23-ton truck (ra = 23 X 103 kg) has its cm located 22 m from the left end of the bridge (point A). Determine the magnitude of the force and type of stress in each strut. [Hint: See Fig. 12-29.]

(Ill) For the Pratt truss shown in Fig. 12-73, determine the force on each member and whether it is tensile or compressive. Assume the truss is loaded as shown, and give results in terms of F. The vertical height is a and each of the four lower horizontal spans has length a.

(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?

The mobile in Fig. 12-74 is in equilibrium. Object B has mass of 0.748 kg. Determine the masses of objects A, C, and D. (Neglect the weights of the crossbars.)

A tightly stretched high wire is 36 m long. It sags 2.1 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?

What minimum horizontal force F is needed to pull a wheel of radius R and mass M over a step of height h as shown in Fig. 12-75 (R > h)l (a) Assume the force is applied at the top edge as shown. (b) Assume the force is applied instead at the wheels center.

A 28-kg round table is supported by three legs equa

When a wood shelf of mass 6.6 kg is fastened inside a slot in a vertical support as shown in Fig. 12-76, 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 slotexplain why). Then calculate (b) the magnitudes of the three forces and (c) the torque exerted by the support (about the left end of the shelf).

A 50-story building is being planned. It is to be 180.0 m high with a base 46.0 m by 76.0 m. Its total mass will be about 1.8 X 107kg, and its weight therefore about 1.8 X 108N. Suppose a 200-km/h wind exerts a force of 950 N/m2 over the 76.0-m-wide face (Fig. 12-77). Calculate the torque about the potential pivot point, the rear edge of the building (where FE acts in Fig. 12-77), and determine whether the building will topple. Assume the total force of the wind acts at the midpoint of the buildings face, and that the building is not anchored in bedrock. [Hint: FE in Fig. 12-77 represents the force that the Earth would exert on the building in the case where the building would just begin to tip.]

The center of gravity of a loaded truck depends on how the truck is packed. If it is 4.0 m . *- high and 2.4 m wide, and its cg is 2.2 m above the ground, how steep a slope can the truck be parked on without tipping over (Fig. 12-78)?

In Fig. 12-79, consider the right-hand (northernmost) section of the Golden Gate Bridge, which has a length di = 343 m. Assume the cg of this span is halfway between the tower and anchor. Determine FX1 and FT2 (which act on the northernmost cable) in terms of mg, the weight of the northernmost span, and calculate the tower height h needed for equilibrium. Assume the roadway is supported only by the suspension cables, and neglect the mass of the cables and vertical wires. [Hint: FX3 does not act on this section.]

Assume that a single-span suspension bridge such as the Golden Gate Bridge has the symmetrical configuration indicated in Fig. 12-79. Assume that the roadway is uniform over the length of the bridge and that each segment of the suspension cable provides the sole support for the roadway directly below it. The ends of the cable are anchored to the ground only, not to the roadway. What must the ratio of d2 to di be so that the suspension cable exerts no net horizontal force on the towers? Neglect the mass of the cables and the fact that the roadway isnt precisely horizontal.

When a mass of 25 kg is hung from the middle of a fixed straight aluminum wire, the wire sags to make an angle of 12 with the horizontal as shown in Fig. 12-80. Determine the radius of the wire.

The forces acting on a 77,000-kg aircraft flying at constant velocity are shown in Fig. 12-81. The engine thrust, Ft = 5.0 X 105 N, acts on a line 1.6 m below the cm. Determine the drag force FD and the distance above the cm that it acts. Assume Fd and Fx are horizontal. (FL is the lift force on the wing.)

A uniform flexible steel cable of weight mg is suspended between two points at the same elevation as shown in Fig. 12-82, where 0 = 56. Determine the tension in the cable (a) at its lowest point, and (b) at the points of attachment, (c) What is the direction of the tension force in each case?

A 20.0-m-long uniform beam weighing 650 N rests on walls A and B, as shown in Fig. 12-83. (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 A and B exert on the beam when the person is standing: (b) at D; (c) at a point 2.0 m to the right of B; (d) 2.0 m to the right of A.

A cube of side I rests on a rough floor. It is subjected to a steady horizontal pull F, exerted a distance h above the floor as shown in Fig. 12-84. As F is increased, the block will either begin to slide, or begin to tip over. Determine the coefficient of static friction /j l s so that (a) 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?]

A 65.0-kg painter is on a uniform 25-kg scaffold supported from above by ropes (Fig. 12-85). There is a 4.0-kg 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. 12-86. His mass m = 68 kg. Determine the normal force exerted by the floor (a) on each hand;(b) on each foot.

A 23-kg sphere rests between two smooth planes as shown in Fig. 12-87. Determine the magnitude of the force acting on the sphere exerted by each plane.

A 15.0-kg ball is supported from the ceiling by rope A. Rope B pulls downward and to the side on the ball. If the angle of A to the vertical is 22 and if B makes an angle of 53 to the vertical (Fig. 12-88), find the tensions in ropes A and B.

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 55 m/s, and that the ultimate strength of body tissue is 5 X 105N/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 2.3 mm in diameter stretches by 0.030% when a mass is suspended from it. How large is the mass?

A 2500-kg trailer is attached to a stationary truck at point B, Fig. 12-89. Determine the normal force exerted by the road on the rear tires at A, and the vertical force exerted on the trailer by the support B.

The roof over a 9.0-m X 10.0-m room in a school has a total mass of 13,600 kg. The roof is to be supported by vertical 2 X 4s (actually about 4.0 cm X 9.0 cm) equally spaced 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 25-kg object is being lifted by pulling on the ends of a 1.15-mm-diameter nylon cord that goes over two 3.00-m-high poles that are 4.0 m apart, as shown in Fig. 12-90. How high above the floor will the object be when the cord breaks?

A uniform 6.0-m-long ladder of mass 16.0 kg leans against a smooth wall (so the force exerted by the wall, Fw, is perpendicular to the wall). The ladder makes an angle of 20.0 with the vertical wall, and the ground is rough.

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 7.8 X 103 kg/m3), and (b) granite (density 2.7 X 103 kg/m3).

A 95,000-kg train locomotive starts across a 280-m-long bridge at time t = 0. The bridge is a uniform beam of mass 23,000 kg and the train travels at a constant 80.0 km/h. What are the magnitudes of the vertical forces, FA(t) and FB(t), on the two end supports, written as a function of time during the trains passage?

A 23.0-kg backpack is suspended midway between two trees by a light cord as in Fig. 12-50. A bear grabs the backpack and pulls vertically downward with a constant force, so that each section of cord makes an angle of 27 below the horizontal. Initially, without the bear pulling, the angle was 15; the tension in the cord with the bear pulling is double what it was when he was not. Calculate the force the bear is exerting on the backpack.

A uniform beam of mass M and length I is mounted on a hinge at a wall as shown in Fig. 12-91. It is held in a horizontal position by a wire making an angle 0 as shown. A mass m is placed on the beam a distance x from the wall, and this distance can be varied. Determine, as a function of x, (a) the tension in the wire and (b) the components of the force exerted by the beam on the hinge.

Two identical, uniform beams are symmetrically set up against each other (Fig. 12-92) on a floor with which they have a coefficient of friction = 0.50. What is the minimum angle the beams can make with the floor and still not fall?

If 35 kg is the maximum mass m that a person can hold in a hand when the arm is positioned with a 105 angle at the elbow as shown in Fig. 12-93, what is the maximum force Fmax that the biceps muscle exerts on the forearm? Assume the forearm and hand have a total mass of 2.0 kg with a cg that is 15 cm from the elbow, and that the biceps muscle attaches 5.0 cm from the elbow.

(a) Estimate the magnitude of the force f m the muscles exert on the back to support the upper body when a person bends forward. Use the model shown in Fig. 12-94b. (b) Estimate the magnitude and direction of the force Fv acting on the fifth lumbar vertebra (exerted by the spine below).

One rod of the square frame shown in Fig. 12-95 contains a turnbuckle which, when turned, can put the rod under tension or compression. If the turnbuckle puts rod AB under a compressive force F, determine the forces produced in the other rods. Ignore the mass of the rods and assume the diagonal rods cross each other freely at the center without friction. [Hint: Use the symmetry of the situation.]

A steel rod of radius R = 15 cm and length 0 stands upright on a firm surface. A 65-kg man climbs atop the rod. (a) Determine the percent decrease in the rods length. (b) When a metal is compressed, each atom throughout its bulk moves closer to its neighboring atom by exactly the same fractional amount. If iron atoms in steel are normally 2.0 X 10-10m apart, by what distance did this interatomic spacing have to change in order to produce the normal force required to support the man? [Note: Neighboring atoms repel each other, and this repulsion accounts for the observed normal force.]

A home mechanic wants to raise the 280-kg engine out of a car. The plan is to stretch a rope vertically from the engine to a branch of a tree 6.0 m above, and back to the bumper (Fig. 12-96). When the mechanic climbs up a stepladder and pulls horizontally on the rope at its midpoint, the engine rises out of the car. (a) How much force must the mechanic exert to hold the engine 0.50 m above its normal position? (b) What is the systems mechanical advantage?

A 2.0-m-high box with a 1.0-m-square base is moved across a rough floor as in Fig. 12-97. The uniform box weighs 250 N and has a coefficient of static friction with the floor of 0.60. What minimum force must be exerted on the box to make it slide? What is the maximum height h above the floor that this force can be applied without tipping the box over? Note f that as the box tips, the normal force and the friction force will act at the lowest corner.

You are on a pirate ship and being forced to walk the plank (Fig. 12-98). You are standing at the point marked C. The plank is nailed onto the deck at point A, and rests on the support 0.75 m away from A. The center of mass of the uniform plank is located at point B. Your mass is 65 kg and the mass of the plank is 45 kg. What is the minimum downward force the nails must exert on the plank to hold it in place?

A uniform sphere of weight mg and radius rQ is tethered to a wall by a rope of length t The rope is tied to the wall a distance h above the contact point of the sphere, as shown in Fig. 12-99. The rope makes an angle 6 with respect to the wall and is not in line with the balls center. The coefficient of static friction between the wall and sphere is /j l . ( a ) Determine the value of the frictional force on the sphere due to the wall. [Hint A wise choice of axis will make this calculation easy.] (b) Suppose the sphere is just on the verge of slipping. Derive an expression for terms of h and 0.

Use the method of joints to determine the force in each member of the truss shown in Fig. 12-100.State whether each member is in tension or compression.

A uniform ladder of mass m and length leans at an angle 0 against a wall, Fig. 12-101. The coefficients of static friction between ladder-ground and ladder-wall are fxG and /itw, respectively. The ladder will be on the verge of slipping when both the static friction forces due to the ground and due to the wall take on their maximum values, (a) Show that the ladder will be stable if 0 > 0mjn, where the minimum 0min is given by tan dmm = ^ ( 1 -MGMw). (b) Leaning ladder problems are often analyzed under the seemingly unrealistic assumption that the wall is frictionless (see Example 12-6). You wish to investigate the magnitude of error introduced by modeling the wall as frictionless, if in reality it is frictional. Using the relation found in part (a), calculate the true value of 0^ for a frictional wall, taking /jlq = piw = 0.40. Then, determine the approximate value of 0min for the frictionless wall model by taking /j lg = 0.40 and /aw = 0. Finally, determine the percent deviation of the approximate value of 0 min from is true value.

In a mountain-climbing technique called the Tyrolean traverse, a rope is anchored on both ends (to rocks or strong trees) across a deep chasm, and then a climber traverses the rope while attached by a sling as in Fig. 12-102. This technique generates tremendous forces in the rope and anchors, so a basic understanding of physics is crucial for safety. A typical climbing rope can undergo a tension force of perhaps 29 kN before breaking, and a safety factor of 10 is usually recommended. The length of rope used in the Tyrolean traverse must allow for some sag to remain in the recommended safety range. Consider a 75-kg climber at the center of a Tyrolean traverse, spanning a 25-m chasm, (a) To be within its recommended safety range, what minimum distance x must the rope sag? (b) If the Tyrolean traverse is set up incorrectly so that the rope sags by only one-fourth the distance I l f / TrT 25 m found in (a), determine the tension in the rope. Will the rope break?

(Ill) A metal cylinder has an original diameter of 1.00 cm and a length of 5.00 cm. A tension test was performed on the specimen and the data are listed in the Table, (a) Graph the stress on the specimen vs. the strain. (b) Considering only the elastic region, find the slope of the best-fit straight line and determine the elastic modulus of the metal.

(Ill) Two springs, attached by a rope, are connected as shown in Fig. 12-103. The length AB is 4.0 m and AC = BC. The spring constant of each spring is k = 20.0 N/m. A force F acts downward at C on the rope. Graph 0 as a function of F from 0 = 0 to 75, assuming the springs are unstretched at 0 = 0.