A 12-ft ladder, weighing 40 lb, leans against a frictionless vertical wall. The lower

Two external shafts of a gearbox carry torques as shown. Determine the vertical components of the forces that must be exerted by the bolts at A and B to maintain the gearbox in equilibrium.

A 2800-kg forklift truck is used to lift a 1500-kg crate. Determine the reaction at each of the two (a) front wheels A, (b) rear wheels B.

A gardener uses a 12-lb wheelbarrow to transport a 50-lb bag of fertilizer. What force must the gardener exert on each handle?

A load of lumber of weight W = 25 kN is being raised as shown by a mobile crane. Knowing that the tension is 25 kN in all portions of cable AEF and that the weight of boom ABC is 3 kN, determine (a) the tension in rod CD, (b) the reaction at pin B.

Three loads are applied as shown to a light beam supported by cables attached at B and D. Neglecting the weight of the beam, determine the range of values of Q for which neither cable becomes slack when P = 0.

Three loads are applied as shown to a light beam supported by cables attached at B and D. Knowing that the maximum allowable tension in each cable is 12 kN and neglecting the weight of the beam, determine the range of values of Q for which the loading is safe when P = 5 kN.

The 10-ft beam AB rests upon, but is not attached, to supports at C and D. Neglecting the weight of the beam, determine the range of values of P for which the beam will remain in equilibrium.

For the beam of Sample Prob. 4.2, determine the range of values of P for which the beam will be safe knowing that the maximum allowable value for each of the reactions is 25 kips and that the reaction at A must be directed upward.

The 40-ft boom AB weighs 2 kips; the distance from the axle A to the center of gravity G of the boom is 20 ft. For the position shown, determine the tension T in the cable and the reaction at A.

The ladder AB, of length L and weight W, can be raised by the cable BC. Determine the tension T required to raise end B just off the floor (a) in terms of W and \(\theta\), (b) if h = 8 ft, L = 10 ft, and W = 35 lb.

Neglecting the radius of the pulley, determine the tension in cable ABD and the reaction at the support C.

The lever AB is hinged at C and attached to a control cable at A. If the lever is subjected at B to a 500-N horizontal force, determine (a) the tension in the cable, (b) the reaction at C.

Determine the reactions at A and B when \(\alpha = 60^\circ\).

The required tension in cable AB is 300 lb. Determine (a) the vertical force P that must be applied to the pedal, (b) the corresponding reaction at C.

Determine the maximum tension that can be developed in cable AB if the maximum allowable magnitude of the reaction at C is 650 lb.

A truss may be supported in three different ways as shown. In each one, determine the reactions at the supports.

A light bar AD is suspended from a cable BE and supports a 20-kg block at C. The extremities A and D of the bar are in contact with frictionless, vertical walls. Determine the tension in cable BE and the reactions A and D.

A light rod, supported by rollers at B, C, and D, is subjected to an 800-N force applied at A. If \(\beta = 0\), determine (a) the reactions at B, C, and D, (b) the rollers that can be safely removed for this loading.

A 160-lb overhead garage door consists of a uniform rectangular panel AC, 84 in. long, supported by the cable AE attached at the middle of the upper edge of the door and by two sets of frictionless rollers at A and B. Each set consists of two rollers located on either side of the door. The rollers A are free to move in horizontal channels, while the rollers B are guided by vertical channels. If the door is held in the position for which BD = 42 in., determine (a) the tension in cable AE, (b) the reaction at each of the four rollers.

In Prob. 4.19, determine the distance BD for which the tension in cable AE is equal to 600 lb.

A 150-kg telephone pole is used to support the ends of two wires as shown. The tension in the wire to the left is 400 N, and, at the point of support, the wire forms an angle of \(10^\circ\) with the horizontal. (a) If the tension \(T_2\) is zero, determine the reaction at the base A. (b) Determine the largest and smallest allowable tension \(T_2\) if the magnitude of the couple at A may not exceed \(900 \ \mathrm{N} \cdot \mathrm{m}\).

The rig shown consists of a 1200-lb horizontal member ABC and a vertical member DBE welded together at B. The rig is being used to raise a 3600-lb crate at a distance x = 12 ft from the vertical member DBE. If the tension in the cable is 4 kips, determine the reaction at E, assuming that the cable is (a) anchored at F as shown in the figure, (b) attached to the vertical member at a point located 1 ft above E.

For the rig and crate of Prob. 4.22, and assuming that the cable is anchored at F as shown, determine (a) the required tension in cable ADCF if the maximum value of the couple at E as x varies from 1.5 to 17.5 ft is to be as small as possible, (b) the corresponding maximum value of the couple.

A traffic-signal pole may be supported in the three ways shown; in part c, the tension in cable BC is to be 1950 N. Determine the reactions for each type of support.

A truss may be supported in eight different ways as shown. All connections consist of frictionless pins, rollers, and short links. In each case, determine whether (a) the truss is completely, partially, or improperly constrained, (b) the reactions are statically determinate or indeterminate, (c) the equilibrium of the truss is maintained in the position shown. Also, wherever possible, compute the reactions, assuming that the magnitude of the force P is 12 kips.

Nine identical rectangular plates, \(500 \times 750 \ \mathrm{mm}\), and each of mass m = 40 kg, are held in a vertical plane as shown. All connections consist of frictionless pins, rollers, and short links. For each case, answer the questions listed in Prob. 4.25, and wherever possible, compute the reactions.

Determine the reactions at B and C when a = 30 mm.

The spanner shown is used to rotate a shaft. A pin fits in a hole at A, while a flat, frictionless surface rests against the shaft at B. If a 300-N force P is exerted on the spanner at D, find the reactions at A and B.

A 10-ft wooden beam weighing 120 lb is supported by a pin and bracket at A and by cable BC. Find the reaction at A and the tension in the cable.

A T-shaped bracket supports a 300-N load as shown. Determine the reactions at A and C when (a) \(\alpha = 90^\circ\), (b) \(\alpha = 45^\circ\).

One end of a rod AB rests in the corner A, and the other is attached to cord BD. If the rod supports a 200-N load at its midpoint C, find the reaction at A and the tension in the cord.

Using the method of Sec. 4.7, solve Prob. 4.12.

Using the method of Sec. 4.7, solve Prob. 4.13.

Using the method of Sec. 4.7, solve Prob. 4.14.

Using the method of Sec. 4.7, solve Prob. 4.15.

Determine the reactions at A and E when \(\alpha = 0\).

Determine (a) the value of \(\alpha\) for which the reaction at A is vertical, (b) the corresponding reactions at A and E.

Determine the reactions at A and B when \(\alpha = 90^\circ\).

Determine the reactions at A and B when \(\alpha = 30^\circ\).

A slender rod BC of length L and weight W is held by two cables as shown. Knowing that cable AB is horizontal and that the rod forms an angle of \(40^\circ\) with the horizontal, determine (a) the angle \(\theta\) that cable CD forms with the horizontal, (b) the tension in each cable.

A slender rod AB of length L and weight W is attached to a collar at A and rests on a small wheel at C. Neglecting the effect of friction and the weight of the collar, determine the angle \(\theta\) corresponding to equilibrium.

Determine the reactions at A and B when a = 7.5 in.

Determine the value of a for which the magnitude of the reaction B is equal to 200 lb.

Rod AB is supported by a pin and bracket at A and rests against a frictionless peg at C. Determine the reactions at A and C when a 170-N vertical force is applied at B.

Solve Prob. 4.44 assuming that the 170-N force applied at B is horizontal and directed to the left.

A uniform plate girder weighing 6000 lb is held in a horizontal position by two crane cables. Determine the angle a and the tension in each cable.

A 12-ft ladder, weighing 40 lb, leans against a frictionless vertical wall. The lower end of the ladder rests on rough ground, 4 ft away from the wall. Determine the reactions at both ends.

A 225-N sign is supported by a pin and bracket at A and by a cable BC. Determine the reaction at A and the tension in the cable.

The L-shaped member ACB is supported by a pin and bracket at C and by an inextensible cord attached at A and B and passing over a frictionless pulley at D. The tension may be assumed to be the same in portions AD and BD of the cord. If the magnitudes of the forces applied at A and B are, respectively, P = 25 lb and Q = 0, determine (a) the tension in the cord, (b) the reaction at C.

For the L-shaped member of Prob. 4.49, (a) express the tension T in the cord in terms of the magnitudes P and Q of the forces applied at A and B, (b) assuming Q = 40 lb, find the smallest allowable value of P if the equilibrium is to be maintained.

Two transmission belts pass over a double-sheaved pulley that is attached to an axle supported by bearings at A and D. The radius of the inner sheave is 125 mm and the radius of the outer sheave is 250 mm. Knowing that when the system is at rest, the tension is 90 N in both portions of belt B and 150 N in both portions of belt C, determine the reactions at A and D. Assume that the bearing at D does not exert any axial thrust.

Solve Prob. 4.51, assuming that the pulley rotates at a constant rate and that \(T_B=104 \mathrm{~N}, T_B^{\prime}=84 \mathrm{~N}\), and \(T_C=175 \mathrm{~N}\).

A \(4 \times 8 \mathrm{ft}\) sheet of plywood weighing 40 lb has been temporarily propped against column CD. It rests at A and B on small wooden blocks and against protruding nails. Neglecting friction at all the surfaces of contact, determine the reactions at A, B and C.

A small wrench is used to raise a 120-lb load. Find (a) the magnitude of the vertical force P that should be applied at C to maintain equilibrium in the position shown, (b) the reactions at A and B, assuming that the bearing at B does not exert any axial thrust.

A 200-mm lever and a 240-mm-diameter pulley are welded to the axle BE that is supported by bearings at C and D. If a 720-N vertical load is applied at A when the lever is horizontal, determine (a) the tension in the cord, (b) the reactions at C and D. Assume that the bearing at D does not exert any axial thrust.

Solve Prob. 4.55 assuming that the axle has been rotated clockwise in its bearings by \(30^\circ\) and that the 720-N load remains vertical.

The rectangular plate shown weighs 80 lb and is supported by three wires. Determine the tension in each wire.

A load W is to be placed on the 80-lb plate of Prob. 4.57. Determine the magnitude of W and the point where it should be placed if the tension is to be 60 lb in each of the three wires.

The 20-kg square plate is supported by the three wires shown. Determine the tension in each wire.

Determine the mass and location of the smallest block that should be placed on the 20-kg plate of Prob. 4.59 if the tensions in the three wires are to be equal.

The 12-ft boom AB is acted upon by the 850-lb force shown. Determine (a) the tension in each cable, (b) the reaction of the ball and socket at A.

Solve Prob. 4.61 assuming that the 850-lb load is applied at point B.

A 7-ft boom is held by a ball and socket at A and by two cables EBF and DC; cable EBF passes around a frictionless pulley at B. Determine the tension in each cable.

A 300-kg crate hangs from a cable that passes over a pulley B and is attached to a support at H. The 100-kg boom AB is supported by a ball and socket at A and by two cables DE and DF. The center of gravity of the boom is located at G. Determine (a) the tension in cables DE and DF, (b) the reaction at A.

The horizontal platform ABCD weighs 60 lb and supports a 240-lb load at its center. The platform is normally held in position by hinges at A and B and by braces CE and DE. If brace DE is removed, determine the reactions at the hinges and the force exerted by the remaining brace CE. The hinge at A does not exert any axial thrust.

A \(1.2 \times 2.4-\mathrm{m}\) sheet of plywood is temporarily held by nails at D and E and by two wooden braces nailed at A, B and C. Wind is blowing on the hidden face of the plywood sheet, and it is assumed that its effect may be represented by a force Pk applied at the center of the sheet. Knowing that each brace becomes unsafe with respect to buckling when subjected to a 1.8-kN axial force, determine (a) the maximum allowable value of the magnitude of P of the wind force, (b) the corresponding value of the z component of the reaction at E. Assume that the nails are loose and do not exert any couple.

A \(3 \times 4-\mathrm{ft}\) plate weighs 150 lb and is supported by hinges at A and B. It is held in the position shown by the 2-ft chain CD. Assuming that the hinge at A does not exert any axial thrust, determine the tension in the chain and the reactions at A and B.

The lid of a roof scuttle weighs 75 lb. It is hinged at corners A and B and maintained in the desired position by a rod CD pivoted at C; a pin at end D of the rod fits into one of several holes drilled in the edge of the lid. For \(\alpha = 50^\circ\), determine (a) the magnitude of the force exerted by rod CD, (b) the reactions at the hinges. Assume that the hinge at B does not exert any axial thrust.

A 10-kg storm window measuring \(900 \times 1500 \ \mathrm{mm}\) is held by hinges at A and B. In the position shown, it is held away from the side of the house by a 600-mm stick CD. Assuming that the hinge at A does not exert any axial thrust, determine the magnitude of the force exerted by the stick and the components of the reactions A and B.

A 20-kg door is made self-closing by hanging a 15-kg counter-weight from a cable attached at C. The door is held open by a force P applied at the knob D in a direction perpendicular to the door. Determine the magnitude of P and the components of the reactions A and B when \(\theta = 90^\circ\). It is assumed that the hinge at A does not exert any axial thrust.

Solve Prob. 4.65 assuming that the hinge at A has been removed and that the hinge at B can exert couples about the axes parallel to the x and y axes, respectively.

Solve Prob. 4.69 assuming that the hinge at A has been removed.

The rigid L-shaped member ABC is supported by a ball and socket at A and three cables. Determine the tension in each cable and the reaction at A caused by the 500-lb load applied at G.

Three rods are welded together to form the “corner” shown. The corner is supported by three smooth eyebolts. Determine the reactions at A, B, and C when P = 1.2 kN, a = 300 mm, b = 200 mm, and c = 250 mm.

The coefficients of friction between the block and the incline are \(\mu_s=0.35\) and \(\mu_k=0.25\). Determine whether the block is in equilibrium, and find the magnitude and direction of the friction force when \(\theta=25^{\circ}\) and P=750 N.

Solve Prob. 4.75 when \(\theta=30^{\circ}\) and P=150 N.

The coefficients of friction between the 50-lb block and the incline are \(\mu_s=0.40\) and \(\mu_k=0.30\). Determine whether the block is in equilibrium, and find the magnitude and direction of the friction force when P=120 lb.

Solve Prob. 4.77 assuming that P = 80 lb.

A support block is acted upon by the two forces shown. Determine the magnitude of P required to start the block up the plane.

Determine the smallest magnitude of the force P that will prevent the support block from sliding down the plane.

Denoting by \(\phi_s\) the angle of static friction between the block and the plane, determine the magnitude and direction of the smallest force P that will cause the block to move up the plane.

A block of mass m = 20 kg rests on a rough plane as shown. Knowing that \(\alpha=25^{\circ}\) and \(\mu_s=0.20\), determine the magnitude and direction of the smallest force P required (a) to start the block up the plane, (b) to prevent the block from moving down the plane.

The coefficients of friction between the block and the rail are \(\mu_s = 0.30\) and \(\mu_k = 0.25\). Knowing that \(\theta = 65^{\circ}\), determine the smallest value of P required (a) to start the block up the rail, (b) to keep it from moving down.

The coefficients of friction between the block and the rail are \(\mu_s = 0.30\) and \(\mu_k = 0.25\). Find the magnitude and direction of the smallest force P required (a) to start the block up the rail, (b) to keep it from moving down.

A 60-kg cabinet is mounted on casters that can be locked to prevent their rotation. The coefficient of static friction between the floor and each caster is 0.35. If h = 600 mm, determine the magnitude of the force P required to move the cabinet to the right (a) if all the casters are locked, (b) if the casters at B are locked and the casters at A are free to rotate, (c) if the casters at A are locked and the casters at B are free to rotate.

A 60-kg cabinet is mounted on casters that can be locked to prevent their rotation. The coefficient of static friction between the floor and each caster is 0.35. Assuming that the casters at both A and B are locked, determine (a) the force P required to move the cabinet to the right, (b) the largest allowable value of h if the cabinet is not to tip over.

A packing crate of mass 40 kg must be moved to the left along the floor without tipping. Knowing that the coefficient of static friction between the crate and the floor is 0.35, determine (a) the largest allowable value of \(\alpha\), (b) the corresponding magnitude of the force P.

A packing crate of mass 40 kg is pulled by a rope as shown. The coefficient of static friction between the crate and the floor is 0.35. If \(\alpha = 40^{\circ}\), determine (a) the magnitude of the force P required to move the crate, (b) whether the crate will slide or tip.

A 180-lb sliding door is mounted on a horizontal rail as shown. The coefficients of static friction between the rail and the door at A and B are 0.20 and 0.30, respectively. Determine the horizontal force that must be applied to the handle C in order to move the door to the left.

Solve Prob. 4.89 assuming that the door is to be moved to the right.

The 10-lb uniform rod AB is held in the position shown by the force P. Knowing that the coefficient of friction is 0.20 at A and B, determine the smallest value of P for which equilibrium is maintained.

In Prob. 4.91, determine the largest value of P for which equilibrium is maintained.

The end A of a slender, uniform rod of length L and weight W bears on the horizontal surface, while its end B is supported by a cord BC. Knowing that the coefficients of friction are \(\mu_s = 0.30\) and \(\mu_k = 0.25\), determine (a) the maximum value of \(\theta\) for which equilibrium is maintained, (b) the corresponding value of the tension in the cord.

Determine whether the rod of Prob. 4.93 is in equilibrium when \(\theta = 30^{\circ}\), and find the magnitude and direction of the friction force exerted on the rod at A.

A slender rod of length L is lodged between peg C and the vertical wall and supports a load P at end A. Knowing that \(L = 12.5a, \theta = 30^{\circ}\), and that the coefficients of friction are \(\mu_s = 0.20\) and \(\mu_k = 0.15\) at C and zero at B, determine whether the rod is in equilibrium.

Solve Prob. 4.95 assuming that \(L = 6a, \theta = 30^{\circ}\), and that the coefficients of friction are \(\mu_s = 0.20\) and \(\mu_k = 0.15\) at B and zero at C.

Find the magnitude of the largest couple M that can be applied to the cylinder if it is not to spin. The cylinder has a weight W and a radius r, and the coefficient of static friction \(\mu_s\) is the same at A and B.

The cylinder has a weight W and a radius r. Express in terms of W and r the magnitude of the largest couple M that can be applied to the cylinder if it is not to spin, assuming that the coefficient of static friction is to be (a) zero at A and 0.35 at B, (b) 0.28 at A and 0.35 at B.

The maximum allowable value for each of the reactions is 150 kN, and the reaction at A must be directed upward. Neglecting the weight of the beam, determine the range of values of P for which the beam is safe.

Determine the reactions at A and B for the loading shown.

The light bar AD is attached to collars B and C that can move freely on vertical rods. Knowing that the surface at A is smooth, determine the reactions at A, B, and C (a) if \(\alpha = 60^{\circ}\), (b) if \(\alpha = 90^{\circ}\).

A movable bracket is held at rest by a cable attached at C and by frictionless rollers at A and B. For the loading shown, determine (a) the tension in the cable, (b) the reactions at A and B.

The 300-lb beam AB carries a 500-lb load at B. The beam is held by a fixed support at A and by the cable CD that is attached to the counterweight W. (a) If W = 1300 lb, determine the reaction at A. (b) Determine the range of values of W for which the magnitude of the couple at A does not exceed \(1500 \ \mathrm{lb} \cdot \mathrm{ft}\).

A 100-kg roller, of diameter 500 mm, is used on a lawn. Determine the force F required to make it roll over a 50-mm obstruction (a) if the roller is pushed as shown, (b) if the roller is pulled as shown.

The overhead transmission shaft AE is driven at a constant speed by an electric motor connected by a flat belt to pulley B. Pulley C may be used to drive a machine tool located directly below C, while pulley D drives a parallel shaft located at the same height as AE. Knowing that \(T_B + T^{\prime}_B = 36 \ \mathrm{lb}, T_C = 40 \ \mathrm{lb}, T^{\prime}_C = 16 \ \mathrm{lb}, T_D = 0\), and \(T^{\prime}_D = 0\), determine (a) the tension in each portion of the belt driving pulley B, (b) the reactions at the bearings A and E caused by the tension in the belts.

A vertical load P is applied at end B of rod BC. The constant of the spring is k and the spring is unstretched when \(\theta = 60^{\circ}\). (a) Neglecting the weight of the rod, express the angle \(\theta\) corresponding to the equilibrium position in terms of P, k, and l. (b) Determine the values of \(\theta\) corresponding to equilibrium if \(P =\frac {1} {4} \mathrm{kl}|).

A force P is applied to a bent rod AD that may be supported in four different ways as shown. In each case determine the reactions at the supports.

A 500-lb marquee, \(8 \times 10 \ \mathrm{ft}\), is held in a horizontal position by two horizontal hinges at A and B and by a cable CD attached to a point D located 5 ft directly above B. Determine the tension in the cable and the components of the reactions at the hinges.

The 10-kg block is attached to link AB and rests on a conveyor belt that is moving to the left. Knowing that the coefficients of friction between the block and the belt are \(\mu_s 0.30\) and \(\mu_k = 0.25\) and neglecting the weight of the link, determine (a) the force in link AB, (b) the horizontal force P that should be applied to the belt to maintain its motion.

A 10-ft uniform plank of weight 45 lb rests on two joists as shown. The coefficient of static friction between the joists and the plank is 0.40. (a) Determine the magnitude of the horizontal force P required to move the plank. (b) Solve part a assuming that a single nail driven into joist A prevents motion of the plank along joist A.