Answer: A beam of length L is carried by three men, one

Figure 12-14 shows three situations in which the same horizontal rod is supported by a hinge on a wall at one end and a cord at its other end. Without written calculation, rank the situations according to the magnitudes of (a) the force on the rod from the cord, (b) the vertical force on the rod from the hinge, and ( c) the horizontal force on the rod from the hinge, greatest first.

In Fig. 12-15, a rigid beam is attached to two posts that are fastened to a floor. A small but heavy safe is placed at the six positions indicated, in turn. Assume that the 1 (3) 2 3 4 5 6 mass of the beam is negligible com- Fig. 12-15 Question 2. pared to that of the safe. (a) Rank the positions according to the force on post A due to the safe, greatest compression first, greatest tension last, and indicate where, if anywhere, the force is zero. (b) Rank them according to the force on post B.

Figure 12-16 shows four overhead views of rotating uniform disks that are sliding across a frictionless floor. Three forces, of (a) (b) (c) (d) Fig. 12-16 Question 3. au ESTIONS 319 (12-24) where !!.xlL is the shearing strain of the object, !!.X is the displacement of one end of the object in the direction of the applied force F (as in Fig. 12-10b), and G is the shear modulus of the object. The stress is FIA. Hydraulic Stress When an object undergoes hydralllic compression due to a stress exerted by a surrounding fluid, Eq. 12-22 is written as !!. V p = B V' (12-25) where p is the pressure (hydraulic stress) on the object due to the fluid, !!. VIV (the strain) is the absolute value of the fractional change in the object's volume due to that pressure, and B is the bulk modulus of the object. magnitude F, 2F, or 3F, act on each disk, either at the rim, at the center, or halfway between rim and center. The force vectors rotate along with the disks, and, in the "snapshots" of Fig. 12-16, point left or right. Which disks are in equilibrium?

A ladder leans against a frictionless wall but is prevented from faIling because of friction between it and the ground. Suppose you shift the base of the ladder toward the wall. Determine whether the following become larger, smaller, or stay the same (in magnitude): (a) the normal force on the ladder from the ground, (b) the force on the ladder from the walI, (c) the static frictional force on the ladder from the ground, and (d) the maximum value t,max of the static frictional force.

Figure 12-17 shows a mobile of toy penguins hanging from a ceiling. Each crossbar is horizontal, has negligible mass, and extends three times as far to the right of the wire supporting it as to the left. Penguin 1 has mass 111\ = 48 kg. What are the masses of (a) penguin 2, (b) penguin 3, and ( c) penguin 4?

Figure 12-18 shows an overhead view of a uniform stick on which four forces act. Suppose we choose a rotation axis through point 0, calculate the torques about that axis due to the forces, and find that these Fig. 12-18 Question 6. torques balance. Will the torques balance if, instead, the rotation axis is chosen to be at (a) point A (on the stick), (b) point B (on line with the stick), or (c) point C (off to one side of the stick)? (d) Suppose, instead, that we find that the torques about point 0 do not balance. Is there another point about which the torques will balance?

In Fig. 12-19, a stationary 5 kg rod AC is held against a wall by a rope and friction between rod and wall. The uniform rod is 1 m long, and angle () = 30. (a) If you are to find the magnitude of the force f on the rod from the rope with a single equation, at what labeled point should a rotation axis be placed? With that choice of axis and counterclockwise torques positive, what is the sign of (b) the torque Til' due to the rod's weight and (c) the torque Tr due to the pull on the rod by the rope? (d) Is the magnitude of T,. greater than, less than, or equal to the magnitude OfT",?

Three pinatas hang from the (stationary) assembly of massless pulleys and cords seen in Fig. 12- 20. One long cord runs from the ceiling at the right to the lower pulley at the left, looping halfway around all the pulleys. Several Fig. 12-19 Question 7. shorter cords suspend pulleys from F the ceiling or pinatas from the pulleys. The weights (in newtons) of two pinatas are given. (a) What is the weight of the third pinata? (Hint: A cord that loops halfway around a pulley pulls on the pulley with a net force that is twice the tension in the cord.) (b) What is the tension in the short cord labeled with T?

In Fig. 12-21, a vertical rod is hinged at its lower end and attached to a cable at its upper end. A horizontal force Fa is to be applied to the rod as shown. If the point at which the force is applied is moved up the rod, does the tension in the cable increase, decrease, or remain the same?

Figure 12-22 shows a horizontal block that is suspended by two wires, A and B, which are identical except for their original lengths. The center of mass of the block is closer to wire B than to wire A. (a) Measuring torques about the block's center of mass, state whether the magnitude of the torque due to wire A is greater than, less than, or equal to the magnitUde of the torque due to wire B. (b) Which wire exerts more force on the block? (c) If the wires are now equal in length, which one was originally shorter (before the P'; A B .com < block was suspended) ?

Figure 12-27 shows a diver of weight 580 N standing at Fig. 12-26 Problem 10. the end of a diving board with a length of L = 4.5 m and negligible mass. The board is fixed to two pedestals (supports) that are separated by distance d = 1.5 m. Of the forces acting on the board, what are the (a) magnitude and (b) direction (up or down) of the force from the left pedestal and the (c) magnitude and (d) direction (up or down) of the force from the right pedestal? Fig. 12-27 Problem 11. (e) Which pedestal (left or right) is being stretched, and (f) which pedestal is being compressed?

In Fig. 12-28, trying to get his car out of mud, a man ties one end of a rope around the front bumper and the other end tightly around a utility pole 18 m away. He then pushes sideways on the rope at its midpoint with a force of 550 N, displacing the center of the rope 0.30 m, but the car barely moves. What is the magnitude of the force on the car from the rope? (The rope stretches somewhat.)

Figure 12-29 shows the anatomical structures in the lower leg and foot that are involved in standing on tiptoe, with the heel raised slightly off the floor so that the foot effec- ~ Calf muscle ~ " Lower leg bones tively contacts the floor only at B point P. Assume distance a = \ "A ;h~ ;;s~~~:~:i:h~ : ~~o;~'r::';~b~ip / ,'7 Of the forces acting on the foot, what are the (a) magni- Fig. 12-29 Problem 13. tude and (b) direction (up or down) of the force at point A from the calf muscle and the (c) magnitude and (d) direction (up or down) of the force at point B from the lower leg bones?

o~, ~;~~n1~~~.0~ !o:~~o~~~~ ~ll.w~~~Ji~l form mass 50.0 kg, is suspended Fig. 12-30 Problem 14. from a building by two cables. The scaffold has dozens of paint cans stacked on it at various points. The total mass of the paint cans is 75.0 kg. The tension in the cable at the right is 722 N. How far horizontally from that cable is the center of mass of the system of paint cans?

Forces Flo F2, and F3 act on the structure of Fig. 12-31, shown in an overhead view. We wish to put the structure in equilibrium by applying a fourth force, at a point such as P. The fourth force has vector components ~, and Fv. We are given that a = 2.0 m, b = 3.0 m,e = 1.0 m,Fj = 20 N,F2 = 10 N, and F3 = 5.0 N. Find (a) F",(b) Fv,and (c) d.

A uniform cubical crate is 0.750 m on each side and weighs 500 N. It rests on a floor with one edge against a very small, fixed obstruction. At what least height above the floor must a horizontal force of magnitude 350 N be applied to the crate to tip it?

In Fig. 12-32, a uniform beam of weight 500 N and length 3.0 m is suspended horizontally. On the left it is hinged to a wall; on the right it is supported by a cable bolted to the wall at distance D above the beam. The least tension that will snap the cable is 1200 N. (a) What value of D corresponds to that tension? (b) To prevent the cable from snapping, should D be increased or decreased from that value?

In Fig. 12-33, horizontal scaffold 2, with uniform mass ,Cable " Fig. 12-32 Problem 17. i'=rr====::::'====ir=='J11 T = ? II p 2 1112 = 30.0 kg and length L2 = 2.00 I-r7TdI L2 171 m, hangs from horizontal scaffold .. ~ ---4 1, with uniform mass /111 = 50.0 Fig. 12-33 Problem 18. kg. A 20.0 kg box of nails lies on scaffold 2, centered at distance d = 0.500 m from the left end. What is the tension T in the cable indicated?

To crack a certain nut in a nutcracker, forces with magnitudes of at least 40 N must act on its shell from both sides. For the nutcracker of Fig. 12-34, with distances L = 12 cm and d = 2.6 cm, what are the force components F~ (perpendicular to the handles) corresponding to that 40 N?

A bowler holds a bowling ball (M = 7.2 kg) in the palm of his hand Fig. 12-34 Problem 19. (Fig. 12-35). His upper arm is vertical; his lower arm (1.8 kg) is horizontal. What is the magnitude of (a) the force of the biceps muscle on the lower arm and (b) the force between the bony structures at the elbow contact point?

The system in Fig. 12-36 is in equilibrium. A concrete block of mass 225 kg hangs from the end of the uniform strut of mass 45.0 kg. For angles 4> = 30.00 and f) = 45.0, find (a) the tension T in the cable and the (b) horizontal and (c) vertical components of the force on the strut from the hinge.

In Fig. 12-37, a 55 kg rock climber is in a lie-back climb along a fissure, with hands pulling on one side of the fissure and feet pressed against the opposite side. The fissure has width w = 0.20 m, and the center of mass of the climber is a horizontal distance d = 0.40 m from the fissure. The coefficient of static friction between hands and T 11 1 rock is ILl = 0.40, and between boots Fig. 12-37 Problem 22. and rock it is fL2 = 1.2. (a) What is the least horizontal pull by the hands and push by the feet that will keep the climber stable? (b) For the horizontal pull of (a), what must be the vertical distance h between hands and feet? If the climber encounters wet rock, so that ILl and fL2 are reduced, what happens to (c) the answer to (a) and (d) the answer to (b)?

In Fig. 12-38, one end of a uniform beam of weight 222 N is hinged to a wall; the other end is supported by a wire that makes angles f) = 30.0 with both wall and beam. Find (a) the tension in the wire and the (b) horizontal and (c) vertical components of the force of the hinge on the beam.

In Fig. 12-39, a climber with a weight of 533.8 N is held by a belay rope connected to her climbing harness and belay device; the force of the rope on her has a line Fig. 12-38 Problem 23. of action through her center of mass. The indicated angles are f) = 40.0 and 4> = 30.0. If her feet are on the verge of sliding on the vertical wall, what is the coefficient of static friction between her climbing shoes and the wall?

In Fig. 12-40, what magnitude of (constant) force F applied horizontally at the axle of the wheel is necessary to raise the wheel over an obstacle of height h = 3.00 cm? The wheel's radius is r = 6.00 cm, and its mass is 111 = 0.800 kg.

In Fig. 12-41, a climber leans out against a vertical ice wall that has negligible friction. Distance a is 0.914 m and distance L is 2.10 m. His center of mass is distance d = 0.940 m from the feet-ground contact point. If he is on the verge of sliding, what is the coefficient of static friction between feet and ground?

In Fig. 12-42, a 15 kg block is )' .. ~ ,\, \> con~~:;~ \0-. ..... d \ held in place via a pulley system. The Fig. 12-41 Problem 26. person's upper arm is vertical; the forearm is at angle () = 30 with the horizontal. Forearm and hand together have a mass of 2.0 kg, with a center of mass at distance dj = 15 cm from the contact point of the forearm bone and the upper-arm bone (humerus). The triceps muscle pulls vertically upward on the forearm at distance d2 = 2.5 cm behind that contact point. Distance d3 is 35 cm. What are the (a) magnitude and (b) direction (up or down) of the force on the forearm from the triceps muscle and the (c) magnitude and (d) direction (up or down) of the force on the forearm from the humerus?

In Fig. 12-43, suppose the length L of the uniform bar is 3.00 m and its weight is 200 N. Also, let the block's weight W = 300 N and the angle () = 30.0. The wire can withstand a maximum tension of 500 N. (a) What is the maximum possible distance x before the wire breaks? A With the block placed at this maximum x, what are the (b) horizontal and (c) vertical components of the force on the bar from the hinge at A?

A door has a height of 2.1 m Fig. 12-43 Problems 28 and 34. along a y axis that extends vertically upward and a width of 0.91 m along an x axis that extends outward from the hinged edge of the door. A hinge 0.30 m from the top and a hinge 0.30 m from the bottom each support half the door's mass, which is 27 kg. In unit-vector notation, what are the forces on the door at (a) the top hinge and (b) the bottom hinge?

In Fig. 12-44, a 50.0 kg uniform square sign, of edge length L = 2.00 m, is hung from a horizontal rod of length d" = 3.00 m and negligible mass. A cable is attached to the end of the rod and to a PROBLEMS 323 point on the wall at distance d" = 4.00 m above the point where the rod is hinged to the wall. (a) What is the tension in the cable? What are the (b) magnitude and (c) direction (left or right) of the horizontal component of the force on the rod from the wall, and the (d) magnitude and (e) direction (up or down) of the vertical component of this force?

In Fig. 12-45, a nonuniform bar is suspended at rest in a horizontal position by two massless cords. One cord makes the angle () = 36.9 with the vertical; the other makes the angle = 53.1 with the vertical. If the Cable Fig. 12-44 Problem 30. length L of the bar is 6.10 m, compute the distance x from the left end of the bar to its center of mass.

In Fig. 12-46, the driver of a car on a horizontal road makes an emergency stop by applying the brakes so that all four wheels lock and skid along the road. The coefficient of kinetic friction between tires and road is 0.40. The separation between the front and rear axles is L = 4.2 m, and the center of mass of the car is located at distance d = 1.8 m behind the front axle and distance h = 0.75 m above the road. The car weighs 11 kN. Find the magnitude of (a) the braking acceleration of the car, (b) the normal force on each rear wheel, (c) the normal force on each front wheel, (d) the braking force on each rear wheel, and (e) the braking force on each front wheel. (Hint: Although the car is not in translational equilibrium, it is in rotational equilibrium.)

Figure 12-47a shows a vertical uniform beam of length L that is hinged at its lower end. A horizontal force ~ is applied to the beam at distance y from the lower end. The beam remains vertical because of a cable attached at the upper end, at angle () with the horizontal. Figure 12-47 b gives the tension T in the cable as a function of the position of the applied force given as a fraction yl L of the beam length. The scale of the T axis is set by T, = 600 N. Figure 12-47c gives the magnitude F" of the horizontal force on the beam from the hinge, also as a function of yiL. Evaluate (a) angle () and (b) the magnitude of ~.

In Fig. 12-43, a thin horizontal bar AB of negligible weight and length L is hinged to a vertical wall at A and supported at B by a thin wire BC that makes an angle () with the horizontal. A block of weight W can be moved anywhere along the bar; its position is defined by the distance x from the wall to its center of mass. As a function of x, find (a) the tension in the wire, and the (b) horizontal and (c) vertical components of the force on the bar from the hinge a t A.

A cubical box is filled with sand and weighs 890 N. We wish to "roll" the box by pushing horizontally on one of the upper edges. (a) What minimum force is required? (b) What minimum coefficient of static friction between box and floor is required? (c) If there is a more efficient way to roll the hox, find the smallest possible force that would have to be applied directly to the box to roll it. (Hint: At the onset of tipping, where is the normal force located?)

Figure 12-48 shows a 70 kg climher hanging by only the crimp hold of one hand on the edge of a shallow horizontal ledge in a rock wall. (The fingers are pressed down to gain purchase.) Her feet touch the rock wall at distance H = 2.0 m directly below her crimped fingers but do not provide any support. Her center of mass is distance a = 0.20 m from the wall. Assume that the force from the ledge supporting her fingers is Fig. 12-48 Problem 36. equally shared by the four fingers. What are the values of the (a) horizontal component F" and (b) vertical component F" of the force on each fingertip?

In Fig. 12-49, a uniform plank, with a length L of 6.10 m and a weight of 445 N, rests on the ground and against a frictionless roller at the top of a wall of height h = 3.05 m. The plank remains in equilibrium for any value of ():::: 70 but slips if () < 70. Find the coefficient of static friction between the plank and the ground.

In Fig. 12-50, uniform beams A and B are attached to a wall with hinges and loosely bolted together (there is no torque of one on the other). Beam A has length LA = 2.40 m and mass 54.0 kg; beam B Fig. 12-49 Problem 37. has mass 68.0 kg. The two hinge points are separated by distance d = 1.80 m. In unit-vector notation, what is the force on (a) beam A due to its hinge, (b) beam A due to the bolt, (c) beam B due to its hinge, and (d) beam B due to the bolt?

For the stepladder shown in Fig. 12-51, sides AC and CE are each 2.44 m long and hinged at C. Bar BD is a tie-rod 0.762 m long, halfway up. A man weighing 854 N climbs l.80 m along the ladder. Assuming that the floor is frictionless and neglecting the mass of the ladder, find (a) the tension in the tie-rod and the magnitudes of the forces on the ladder from the floor at (b) A and (c) E. (Hint: Isolate parts of the ladder in applying the equilibrium conditions.)

Figure 12-52a shows a horizon- it E tal uniform beam of mass I11b and length L that is supported on the left by Fig. 12-51 Problem 39. a hinge attached to a wall and on the right by a cable at angle o with the horizontal. A package of mass Il1p is positioned on the beam at a distance x from the left end. 111e total mass is I11b + I11p = 61.22 kg. Figure 12-52b gives the tension T in the cable as a function of the package's position given as a fraction xlL of the beam length. The scale of the T axis is set by Ta = 500 Nand Tb 700 N. Evaluate (a) angle 0, (b) mass I11b, and (c) mass I11p.

A crate, in the form of a cube with edge lengths of 1.2 m, contains a piece of machinery; the center of mass of the crate and its contents is located 0.30 m above the crate's geometrical center. The crate rests on a ramp that makes an angle 0 with the horizontal. As 0 is increased from zero, an angle will be reached at which the crate will either tip over or start to slide down the ramp. If the coefficient of static friction /Ls between ramp and crate is 0.60, (a) does the crate tip or slide and (b) at what angle 0 does this occur? If /Ls = 0.70, (c) does the crate tip or slide and (d) at what angle 0 does this occur? (Hint: At the onset of tipping, where is the normal force located?)

In Fig.12-5 and the associated sample problem, let the coefficient of static friction /Ls between the ladder and the pavement be 0.53. How far (in percent) up the ladder must the firefighter go to put the ladder on the verge of sliding?

A horizontal aluminum rod 4.8 cm in diameter projects 5.3 cm from a wall. A 1200 kg object is suspended from the end of the rod. The shear modulus of aluminum is 3.0 X 1010 N/m2 Neglecting the rod's mass, find (a) the shear stress on the rod and (b) the vertical deflection of the end of the rod.

Figure 12-53 shows the stress-strain curve for a material. ~S "- The scale of the stress axis is set by !- s = 300, in units of 106 N/m2 What ;3 are (a) the Young's modulus and (b) '" the approximate yield strength for [fJ ~ this material?

In Fig. 12-54, a lead brick rests horizontally on cylinders A and B. The areas of the top faces of the cylinders are related by AA = 2AB; the Young's moduli of the cylinders are related by EA = 2EB The cylinders had identical lengths before the brick was placed on them. What fraction of the brick's mass is supported (a) by cylinder A and (b) by cylinder B? The horizontal distances between the center of mass of the brick and the centerlines of the cylinders are dA for cylinder Fig. 12-53 Problem 44. A and dB for cylinder B. (c) What is the ratio dA/dB?

Figure 12-55 shows an approximate plot of stress versus strain for a spider-web thread, out to the point of breaking at a strain of 2.00. The vertical axis scale is set by values a = 0.12 GN/m2, b = 0.30 GN/m2, and c = 0.80 GN/m2. Assume that the thread has an initiallength of 0.80 cm, an initial cross-sectional area of 8.0 X 10- 12 m2, and (during stretching) a constant volume. Assume also that when the single thread snares a flying insect, the insect's kinetic energy is transferred to the stretching of the thread. (a) HilI\' much kinetic energy would put the thread on the verge of breaking? What is the kinetic energy of (b) a fruit fly of mass 6.00 mg and speed 1.70 mls and (c) a bumble bee of mass 0.388 g and speed 0.420 m/s? Would (d) the fruit fly and (e) the bumhle bee break the thread?

A tunnel of length L = 150 m, height H = 7.2 m, and width 5.8 m (with a flat roof) is to be constructed at distance d = 60 m beneath the ground. (See Fig. 12-56.) The tunnel roof is to be supported entirely by square steel columns, each with a cross-sectional area of 960 cm2 The mass of 1.0 cm3 of the ground material is 2.8 g. (a) What is the total weight of the ground material the columns 111lLiL support? (b) How many columns are needed to keep the compre:;- sive stress on each column at one-half its ultimate strength?

Figure 12-57 shows the stress versus strain plot for an aluminum wire that is stretched by a machine pulling in opposite directions at the two ends of the wire. The scale of the stress axis is set by s = 7.0, in units of 107 N/m2 The wire has an initial length of 0.800 m and an initial cross-sectional area of 2.00 X 10-6 m2. How much work does the force from the machine do on the wire to produce a strain of 1.00 X 1O-3?

In Fig. 12-58, a 103 kg uniform log hangs by two steel wires, A and B, both of radius 1.20 mm. Initially, wire A was 2.50 m long and 2.00 mm shorter than wire B. The log is now horizontal. What are the magnitudes of the forces on it from (a) wire A and (b) wire B? (c) What is the ratio dAlds?

Figure 12-59 represents an insect caught at the midpoint of a spider-web thread. The thread breaks under a stress of 8.20 X 108 N/m2 and a strain of 2.00. Initially, it was horizontal and had a length of 2.00 cm and a crosssectional area of 8.00 X 10- 12 m2 As the thread was stretched under the weight of the insect, its volume remained constant. If the weight of the insect puts the thread on the verge of breaking, what is the insect's mass? (A spider's web is built to break if a potentially harmful insect, such as a bumble bee, becomes snared in the web.)

Figure 12-60 is an overhead view of a rigid rod that turns about a vertical axle until the identical rubber stoppers A and B are forced against rigid walls at distances I'A = 7.0 cm and I'B = 4.0 cm from the axle. Initially the stoppers touch the walls without being compressed. Then force F of magnitude 220 N is applied perpendicular to the rod at a distance R = 5.0 cm from the axle. Find the magnitude of the force compressing (a) stopper A and (b) stopper B.

After a fall, a 95 kg rock climber finds himself dangling from the end of a rope that had been 15 m long and 9.6 mrn in diameter but has stretched by 2.8 cm. For the rope, calculate (a) the strain, (b) the stress, and (c) the Young's modulus.

In Fig. 12-61, a rectangular slab of slate rests on a bedrock surface inclined at angle e = 26. The slab has length L = 43 m, thickness T = 2.5 m, and width W = 12 m, and 1.0 cm3 of it has a mass of 3.2 g. The coefficient of static friction between slab and bedrock is 0.39. (a) Fig. 12-61 Problem 53. Calculate the component of the gravitational force on the slab parallel to the bedrock surface. (b) Calculate the magnitude of the static frictional force on the slab. By comparing (a) and (b), you can see that the slab is in danger of sliding. This is prevented only by chance protrusions of bedrock. (c) To stabilize the slab, bolts are to be driven perpendicular to the bedrock surface (two bolts are shown). If each bolt has a crosssectional area of 6.4 cm2 and will snap under a shearing stress of 3.6 X 108 N/m2, what is the minimum number of bolts needed? Assume that the bolts do not affect the normal force.

A uniform ladder whose length is 5.0 m and whose weight is 400 N leans against a frictionless vertical wall. The coefficient of static friction between the level ground and the foot of the ladder is 0.46. What is the greatest distance the foot of the ladder can be placed from the base of the wall without the ladder immediately slipping?

In Fig. 12-62, block A (mass 10 kg) is in equilibrium, but it would slip if block B (mass 5.0 kg) were any heavier. For angle e = 30, what is the coefficient of static friction between block A and the surface below it?

Figure 12-63a shows a uniform ramp between two buildings that allows for motion between the buildings due to strong winds. At its left end, it is hinged to the building wall; at its right end, it has a roller that A can roll along the building wall. B There is no vertical force on the roller from the building, only a horizontal force with magnitude F". The horizontal distance be- Fig. 12-62 Problem 55. tween the buildings is D = 4.00 m. The rise of the ramp is h = 0.490 m. A man walks across the ramp from the left. Figure 12-63b gives F" as a function of the horizontal distance x of the man from the building at the left. The scale of the F" axis is set by a = 20 kN and b = 25 kN. What are the masses of (a) the ramp and (b) the man?

In Fig. 12-64, a 10 kg sphere is supported on a frictionless plane inclined at angle e = 45 from the horizontal. Angle is 25. Calculate the tension in the cable.

In Fig. 12-65a, a uniform 40.0 kg beam is centered over two rollers. Vertical lines across the beam mark off equal lengths. Two of the lines are centered over the rollers; a 10.0 kg package of tamales is centered over roller B. What are the magnitudes of the forces on the beam from (a) roller A and (b) roller B? The beam is then rolled to the left until the right-hand end is centered over roller B (Fig. 12-65b). What now are the magnitudes of the forces on the beam from (c) roller A and (d) roller B? Next, the beam is rolled Fig. 12-64 Problem 57. (a) (b) to the right. Assume that it has a Fig. 12-65 Problem 58. length of 0.800 m. (e) What horizontal distance between the package and roller B puts the beam on the verge of losing contact with roller A?

In Fig. 12-66, an 817 kg construction bucket is suspended by a cable A that is attached at 0 to two other cables Band C, making angles 81 = 51.0 and 82 = 66.0 with the horizontal. Find the ten- )1 sions in (a) cable A, (b) cable B, and (c) cable C. (Hint: To avoid solving two equations in two unknowns, position the axes as shown in the figure.)

In Fig. 12-67, a package of mass m hangs from a short cord that is tied to the wall via cord 1 and to the ceiling via cord 2. Cord 1 is at angle = 40 with the horizontal; cord 2 is at angle 8. (a) For what value of 8 is the tension in cord 2 minimized? (b) In terms of mg, what is the minimum tension in cord 2?

The force F in Fig. 12-68 keeps the 6.40 kg block and the pulleys in equilibrium. The pulleys x / Fig. 12-66 Problem 59. have negligible mass and friction. Fig. 12-67 Problem 60. Calculate the tension T in the upper cable. (Hint: When a cable wraps halfway around a pulley as here, the magnitude of its net force on the pulley is twice the tension in the cable.)

A mine elevator is supported by a single steel cable 2.5 cm in diameter. The total mass of the elevator cage and occupants is 670 kg. By how much does the cable stretch when the elevator hangs by (a) 12 m of cable and (b) 362 m of cable? (Neglect the mass of the cable.)

Four bricks of length L, identical and uniform, are stacked on top of one another (Fig. 12-69) in such a way that part of each extends beyond the one beneath. Find, in terms of L, the maximum values of (a) ab (b) Fig. 12-68 Problem 61. a2, ( c) a3, (d) a4, and (e) h, such that the stack is in equilibrium.

In Fig. 12-70, two identical, uniform, and frictionless spheres, each of mass 111, rest in a rigid rectangular container. A line connecting their centers is at 45 to the horizontal. Find the magnitudes of the forces on the spheres from (a) the bottom ofthe container, (b) the left side of the container, (c) the right side of the container, and (d) each other. (Hint: The force of one sphere on the other is directed along the center-center line.)

In Fig. 12-71, a uniform beam with a weight of 60 N and a length of 3.2 m is hinged at its lower end, and a horizontal force F of magnitude 50 N acts at its upper end. The beam is held vertical by a cable that makes angle 8 = 25 with the ground and is attached to the beam at height h = 2.0 m. What are (a) the tension in the cable and (b) the force on the beam from the hinge in unit-vector notation?

A uniform beam is 5.0 m long and has a mass of 53 kg. In Fig. 12-72, the beam is supported in a horizontal position by a hinge and a cable, with angle 8 = 60. In unit-vector notation, what is the force on the beam from the hinge?

A solid copper cube has an edge length of 85.5 cm. How much stress must be applied to the cube to reduce the edge length to 85.0 cm? The bulk modulus of copper is 1.4 X 1011 N/m2

A construction worker attempts to lift a uniform beam off the floor and raise it to a vertical position. The beam is 2.50 m long and weighs 500 N. At a certain instant the worker holds the beam momentarily at rest with one end at distance d = 1.50 m above the floor, as shown in Fig. 12-73, by exerting a force P on the beam, perpendicular to the beam. (a) What is the magnitude P? (b) What is the magnitude of the (net) force of the floor on the beam? (c) What is the minimum value the coefficient of static friction between beam and floor can have in order for the beam not to slip at this instant?

In Fig. 12-74, a uniform rod of mass ~' 111 is hinged to a building at its lower end, while its upper end is held in place by a rope attached to the wall. If angle Bj = 60, what value must angle B2 have so that the tension in the rope is equal to 111g/2?

A 73 kg man stands on a level bridge of length L. He is at distance Ll4 from one end. The bridge is uniform and weighs 2.7 kN. What are the magnitudes of the vertical forces on the Fig. 12-74 Problem 69. bridge from its supports at (a) the end farther from him and (b) the nearer end?

A uniform cube of side length 8.0 em rests on a horizontal floor. The coefficient of static friction between cube and floor is p.,. A horizontal pull P is applied perpendicular to one of the vertical faces of the cube, at a distance 7.0 em above the floor on the vertical midline of the cube face. The magnitude of P is gradually increased. During that increase, for what values of p., will the cube eventually (a) begin to slide and (b) begin to tip? (Hint: At the onset of tipping, where is the normal force located?)

The system in Fig. 12-75 is in equilibrium. The angles are Bj = 60 and B2 = 20, and the ball has mass M = 2.0 kg. What is the tension in (a) string ab and (b) string be?

A uniform ladder is 10 m long and weighs 200 N. In Fig. 12-76, the ladder leans against a vertical, frictionless wall at height h = 8.0 m above the ground. A horizontal force F is applied to the ladder at distance d = 2.0 m from its base (measured along the ladder). (a) If force magnitUde F = 50 N, what is the force of the ground on the ladder, in unit-vector notation? (b) If F = 150 N, what is the force of the ground on the ladder, also in unit-vector notation? (c) Suppose the coefficient of y L.\. 11 Fig. 12-76 Problem 73. static friction between the ladder and the ground is 0.38; for what minimum value of the force magnitude F will the base of the ladder just barely start to move toward the wall?

A pan balance is made up of a rigid, massless rod with a hanging pan attached at each end. The rod is supported at and free to rotate about a point not at its center. It is balanced by unequal masses placed in the two pans. When an unknown mass 111 is placed in the left pan, it is balanced by a mass 1111 placed in the right pan; when the mass 111 is placed in the right pan, it is balanced by a mass 1112 in the left pan. Show that 111 = Y11111112'

The rigid square frame in Fig. T T 12-77 consists of the four side bars AB, BC, CD, and DA plus two diagonal bars AC and BD, which pass each other freely at E. By means of the turnbuckle G, bar AB is put under tension, as if its ends were subject to horizontal, outward forces T of magnitude 535 N. (a) Which of the other bars are in ten- Fig. 12-77 Problem 75. sion? What are the magnitudes of (b) the forces causing the tension in those bars and (c) the forces causing compression in the other bars? (Hint: Symmetry considerations can lead to considerable simplification in this problem.)

A gymnast with mass 46.0 kg stands on the end of a uniform balance beam as shown in Fig. 12-78. The beam is 5.00 m long and has a mass of 250 kg (excluding the mass of the two supports). Each support is 0.540 m from its end of the beam. In unit-vector notation, what are the forces on the beam due to (a) support 1 and (b) support 2?

Figure 12-79 shows a 300 kg cylinder Three steel wires support the cylinder from a ceiling. Wires 1 and 3 are attached at the ends of the cylinder, and wire 2 is attached at the center. The wires each have a cross-sectional area that is horizontal. rCeiling of 2.00 X 10- 6 m2 Initially (be- Fig. 12-79 Problem 77. fore the cylinder was put in place) wires 1 and 3 were 2.0000 m long and wire 2 was 6.00 mm longer than that. Now (with the cylinder in place) all three wires have been Cable stretched. What is the tension in (a) wire 1 and (b) wire 2?

In Fig. 12-80, a uniform beam of length 12.0 m is supported by a horizontal cable and a hinge at angle B = 50.0. The tension in the cable is 400 N. In unit-vector notation, what are (a) the gravitational force on the beam and (b) the force on the beam from the hinge?

Four bricks of length L, identical and uniform, are stacked on a table in two ways, as shown in Fig. 12-81 (compare with Problem 63). We seek to maximize the overhang distance h in both arrangements. Find the optimum distances at. a2, bj, and b2, and calculate h for the two arrangements.

A cylindrical aluminum rod, with an initial length of 0.8000 m and radius 1000.0 pm, is clamped in place at one end and then stretched by a machine pulling parallel to its length at its other end. Assuming that the rod's density (mass per unit volume) does not change, find the force magnitude that is required of the PROBLEMS 329 machine to decrease the radius to 999.9 /Lm. (The yield strength is not exceeded.)

A beam of length L is carried by three men, one man at one end and the other two supporting the beam between them on a crosspiece placed so that the load of the beam is equally divided among the three men. How far from the beam's free end is the crosspiece placed? (Neglect the mass of the crosspiece.)

If the (square) beam in Fig. 12-6a and the associated sample problem is of Douglas fir, what must be its thickness to keep the compressive stress on it to k of its ultimate strength?

Figure 12-82 shows a stationary arrangement of two crayon boxes and three cords. Box A has a mass of 11.0 kg and is on a ramp at angle e = 30.0; box B has a mass of 7.00 kg and hangs on a cord. The cord connected to box A is parallel to the ramp, which is frictionless. (a) What is the tension in the upper cord, and (b) what angle does that cord make with the horizontal?