?Figure 12-59 shows the stress versus strain plot for an aluminum wire that is stretched

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 \(3.0 × 10^{10} N/m^{2}\). Neglecting the rod’s mass, find (a) the shear stress on the rod and (b) the vertical deflection of the end of the rod. Text Transcription: 3.0 × 10^10 N/m^2

Figure 12-55 shows the stress–strain curve for a material. The scale of the stress axis is set by s = 300, in units of \(10^{6} N/m^{2}\). What are (a) the Young’s modulus and (b) the approximate yield strength for this material? Text Transcription: 10^{6} N/m^{2}

In Fig. 12-56, a lead brick rests horizontally on cylinders A and B. The areas of the top faces of the cylinders are related by \(A_{A} = 2A_{B}\); the Young’s moduli of the cylinders are related by \(E_{A} = 2E_{B}\). 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 \(d_{A}\) for cylinder A and \(d_{B}\) for cylinder B. (c) What is the ratio \(d_{A}/d_{B}\)? Text Transcription: A_A = 2A_B E_A = 2E_B d_A d_B d_A/d_B

Figure 12-57 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 \(a = 0.12 GN/m^{2}\), \(b = 0.30 GN/m^{2}\), and \(c = 0.80 GN/m^{2}\). Assume that the thread has an initial length of 0.80 cm, an initial cross-sectional area of \(8.0 × 10^{?12} m^{2}\), and (during stretching) a constant volume. The strain on the thread is the ratio of the change in the thread’s length to that initial length, and the stress on the thread is the ratio of the collision force to that initial cross-sectional area. Assume that the work done on the thread by the collision force is given by the area under the curve on the graph. 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) How 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 m/s and (c) a bumble bee of mass 0.388 g and speed 0.420 m/s? Would (d) the fruit fly and (e) the bumble bee break the thread? Text Transcription: a = 0.12 GN/m^2 b = 0.30 GN/m^2 c = 0.80 GN/m^2 8.0 × 10^?12 m^2

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. 12-58.) The tunnel roof is to be supported entirely by square steel columns, each with a cross-sectional area of \(960 cm^{2}\). The mass of \(1.0 cm^{3}\) of the ground material is 2.8 g. (a) What is the total weight of the ground material the columns must support? (b) How many columns are needed to keep the compressive stress on each column at one-half its ultimate strength? Text Transcription: 960 cm^2 1.0 cm^3

Figure 12-59 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 \(10^{7} N/m^{2}\). The wire has an initial length of 0.800 m and an initial cross sectional area of \(2.00 × 10^{?6} m^{2}\). How much work does the force from the machine do on the wire to produce a strain of \(1.00 × 10^{?3}\)? Text Transcription: 10^7 N/m^2 2.00 × 10^?6 m^2 1.00 × 10^?3

In Fig. 12-60, 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 \(d_{A}/d_{B}\_? Text Transcription: d_A/d_B

Figure 12-61 represents an insect caught at the midpoint of a spider-web thread. The thread breaks under a stress of \(8.20 × 10^{8} N/m^{2}\) and a strain of 2.00. Initially, it was horizontal and had a length of 2.00 cm and a cross sectional area of \(8.00 × 10^{?12} m^{2}\). 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.) Text Transcription: 8.20 × 10^8 N/m^2 8.00 × 10^?12 m^2

Figure 12-62 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 \(r^{A} = 7.0 cm\) and \(r^{B} = 4.0 cm\) from the axle. Initially the stoppers touch the walls without being compressed. Then force \(\vec F\) 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. Text Transcription: r_A = 7.0 cm r_B = 4.0 cm vec F

In Fig. 12-63, a rectangular slab of slate rests on a bedrock surface inclined at angle ? = 26°. The slab has length L = 43 m, thickness T = 2.5 m, and width W = 12 m, and \(1.0 cm^{3}\) of it has a mass of 3.2 g. The coefficient of static friction between slab and bedrock is 0.39. (a) 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 cross-sectional area of \(6.4 cm^{2}\) and will snap under a shearing stress of \(3.6 × 10^{8} N/m^{2}\), what is the minimum number of bolts needed? Assume that the bolts do not affect the normal force. Text Transcription: 1.0 cm^3 6.4 cm^2 3.6 × 10^8 N/m^2

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?

Figure 12-65a 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 can roll along the building wall. There is no vertical force on the roller from the building, only a horizontal force with magnitude \(F_{h}\). The horizontal distance between 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 12-65b gives \(F_{h}\) as a function of the horizontal distance x of the man from the building at the left. The scale of the \(F_{h}\) axis is set by a = 20 kN and b = 25 kN. What are the masses of (a) the ramp and (b) the man? Text Transcription: F_h

In Fig. 12-66, a 10 kg sphere is supported on a frictionless plane inclined at angle ? = 45° from the horizontal. Angle ???? is 25°. Calculate the tension in the cable.

In Fig. 12-67a, 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. 12-67b). What now are the magnitudes of the forces on the beam from (c) roller A and (d) roller B? Next, the beam is rolled to the right. Assume that it has a 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. 12-68, an 817 kg construction bucket is suspended by a cable A that is attached at O to two other cables B and C, making angles \(?_{1} = 51.0°\) and \(?_{2} = 66.0°\) with the horizontal. Find the tensions 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.) Text Transcription: ?_1 = 51.0° ?_2 = 66.0°

In Fig. 12-69, 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 ?. (a) For what value of ? is the tension in cord 2 minimized? (b) In terms of mg, what is the minimum tension in cord 2?

The force \(\vec F\) in Fig. 12-70 keeps the 6.40 kg block and the pulleys in equilibrium. The pulleys have negligible mass and friction. 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.) Text Transcription: vec F

Four bricks of length L, identical and uniform, are stacked on top of one another (Fig. 12-71) in such a way that part of each extends beyond the one beneath. Find, in terms of L, the maximum values of (a) \(a_{1}\), (b) \(a_{2}\), (c) \(a_{3}\), (d) \(a_{4}\), and (e) h, such that the stack is in equilibrium, on the verge of falling. Text Transcription: a_1 a_2 a_3 a_4

In Fig. 12-72, two identical, uniform, and frictionless spheres, each of mass m, 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 of the 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. 12-73, 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 \(\vec F\) of magnitude 50 N acts at its upper end. The beam is held vertical by a cable that makes angle ? = 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. 12-74, the beam is supported in a horizontal position by a hinge and a cable, with angle ? = 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 \(1.4 × 10^{11} N/m^{2}\). Text Transcription: 1.4 × 10^11 N/m^2

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. 12-75, by exerting a force \(\vec P\) 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? Text Transcription: vec P

In Fig. 12-76, a uniform rod of mass m 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 \(?_{1} = 60°\), what value must angle \(?_{2}\) have so that the tension in the rope is equal to mg/2? Text Transcription: ?_1 = 60° ?_2

A uniform cube of side length 8.0 cm rests on a horizontal floor. The coefficient of static friction between cube and floor is ????. A horizontal pull \(\vec P\) is applied perpendicular to one of the vertical faces of the cube, at a distance 7.0 cm above the floor on the vertical midline of the cube face. The magnitude of \(\vec P\) is gradually increased. During that increase, for what values of ???? 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?) Text Transcription: vec P

The system in Fig. 12-77 is in equilibrium. The angles are \(?_{1} = 60°\) and \(?_{2} = 20°\), and the ball has mass M = 2.0 kg. What is the tension in (a) string ab and (b) string bc? Text Transcription: ?_1 = 60° ?_2 = 20°

A uniform ladder is 10 m long and weighs 200 N. In Fig. 12-78, the ladder leans against a vertical, frictionless wall at height h = 8.0 m above the ground. A horizontal force \(\vec F\) 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 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? Text Transcription: vec F

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 m is placed in the left pan, it is balanced by a mass \(m_{1}\) placed in the right pan; when the mass m is placed in the right pan, it is balanced by a mass \(m_{2}\) in the left pan. Show that \(m=\sqrt{m_{1} m_{2}}\). Text Transcription: m_1 m_2 m = sqrt m_1 m_2

The rigid square frame in Fig. 12-79 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 \(\vec T\) of magnitude 535 N. (a) Which of the other bars are in tension? 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.) Text Transcription: vec T

A gymnast with mass 46.0 kg stands on the end of a uniform balance beam as shown in Fig. 12-80. 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 12-81 shows a 300 kg cylinder that is horizontal. 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 of \(2.00 × 10^{?6} m^{2}\). Initially (before 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 stretched. What is the tension in (a) wire 1 and (b) wire 2? Text Transcription: 2.00 × 10^?6 m^2

In Fig. 12-82, a uniform beam of length 12.0 m is supported by a horizontal cable and a hinge at angle ? = 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. 12-83 (compare with Problem 63). We seek to maximize the overhang distance h in both arrangements. Find the optimum distances \(a_{1}, a_{2}, b_{1}\), and \(b_{2}\), and calculate h for the two arrangements. Text Transcription: a_1, a_2, b_1 b_2

Figure 12-84 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 ? = 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?

A particle is acted on by forces given, in newtons, by \(\vec F_{1} = 8.40 \hat{\mathrm{i}}-5.70 \hat{\mathrm{j}}\) and \(\vec{F}_{2}=16.0 \hat{\mathrm{i}}+4.10 \hat{\mathrm{j}}\). (a) What are the x component and (b) y component of the force \(\vec F_{3}\) that balances the sum of these forces? (c) What angle does \(\vec F_{3}\) have relative to the +x axis? Text Transcription: vec F_1 = 8.40 hat i ? 5.70 hat j vec F_2 = 16.0 hat i + 4.10 hat j vec F_3

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 mm 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.

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?

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.)

A 73 kg man stands on a level bridge of length L. He is at distance L/4 from one end. The bridge is uniform and weighs 2.7 kN. What are the magnitudes of the vertical forces on the bridge from its supports at (a) the end farther from him and (b) the nearer end?

A cylindrical aluminum rod, with an initial length of 0.8000 m and radius 1000.0 ????m, 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 machine to decrease the radius to 999.9 ????m. (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 16 of its ultimate strength?

A makeshift swing is constructed by making a loop in one end of a rope and tying the other end to a tree limb. A child is sitting in the loop with the rope hanging vertically when the child’s father pulls on the child with a horizontal force and displaces the child to one side. Just before the child is released from rest, the rope makes an angle of 15° with the vertical and the tension in the rope is 280 N. (a) How much does the child weigh? (b) What is the magnitude of the (horizontal) force of the father on the child just before the child is released? (c) If the maximum horizontal force the father can exert on the child is 93 N, what is the maximum angle with the vertical the rope can make while the father is pulling horizontally?

A trap door in a ceiling is 0.91 m square, has a mass of 11 kg, and is hinged along one side, with a catch at the opposite side. If the center of gravity of the door is 10 cm toward the hinged side from the door’s center, what are the magnitudes of the forces exerted by the door on (a) the catch and (b) the hinge?

The leaning Tower of Pisa is 59.1 m high and 7.44 m in diameter. The top of the tower is displaced 4.01 m from the vertical. Treat the tower as a uniform, circular cylinder. (a) What additional displacement, measured at the top, would bring the tower to the verge of toppling? (b) What angle would the tower then make with the vertical?