A dirigible is tethered by a cable attached to its cabin

A 20-lb force is applied to the control rod AB as shown. Knowing that the length of the rod is 9 in. and that \(a=25^{\circ}\), determine the moment of the force about point B by resolving the force into horizontal and vertical components.

A 20-lb force is applied to the control rod AB as shown. Knowing that the length of the rod is 9 in. and that \(a=25^{\circ}\), determine the moment of the force about point B by resolving the force into components along AB and in a direction perpendicular to AB.

A 20-lb force is applied to the control rod AB as shown. Knowing that the length of the rod is 9 in. and the moment of the force about B is \(120\mathrm{\ lb}\cdot\mathrm{in.}\) clockwise, determine the value of a.

A crate of mass 80 kg is held in the position shown. Determine (a) the moment produced by the weight W of the crate about E, (b) the smallest force applied at B that creates a moment of equal magnitude and opposite sense about E.

A crate of mass 80 kg is held in the position shown. Determine (a) the moment produced by the weight W of the crate about E, (b) the smallest force applied at A that creates a moment of equal magnitude and opposite sense about E, (c) the magnitude, sense, and point of application on the bottom of the crate of the smallest vertical force that creates a moment of equal magnitude and opposite sense about E.

A 300-N force P is applied at point A of the bell crank shown. (a) Compute the moment of the force P about O by resolving it into horizontal and vertical components. (b) Using the result of part a, determine the perpendicular distance from O to the line of action of P.

A 400-N force P is applied at point A of the bell crank shown. (a) Compute the moment of the force P about O by resolving it into components along line OA and in a direction perpendicular to that line. (b) Determine the magnitude and direction of the smallest force Q applied at B that has the same moment as P about O.

It is known that a vertical force of 200 lb is required to remove the nail at C from the board. As the nail first starts moving, determine (a) the moment about B of the force exerted on the nail, (b) the magnitude of the force P that creates the same moment about B if \(a=10^{\circ}\), (c) the smallest force P that creates the same moment about B.

It is known that the connecting rod AB exerts on the crank BC a 500-lb force directed down and to the left along the centerline of AB. Determine the moment of the force about C.

It is known that the connecting rod AB exerts on the crank BC a 500-lb force directed down and to the left along the centerline of AB. Determine the moment of the force about C.

A winch puller AB is used to straighten a fence post. Knowing that the tension in cable BC is 1040 N and length d is 1.90 m, determine the moment about D of the force exerted by the cable at C by resolving that force into horizontal and vertical components applied (a) at point C, (b) at point E.

It is known that a force with a moment of \(960\mathrm{\ N}\cdot\mathrm{m}\) about D is required to straighten the fence post CD. If d = 2.80 m, determine the tension that must be developed in the cable of winch puller AB to create the required moment about point D.

It is known that a force with a moment of \(960\mathrm{\ N}\cdot\mathrm{m}\) about D is required to straighten the fence post CD. If the capacity of winch puller AB is 2400 N, determine the minimum value of distance d to create the specified moment about point D.

A mechanic uses a piece of pipe AB as a lever when tightening an alternator belt. When he pushes down at A, a force of 485 N is exerted on the alternator at B. Determine the moment of that force about bolt C if its line of action passes through O.

Form the vector products \(\mathbf{B} \times \mathbf{C}\) and \(\mathbf{B}^{\prime} \times \mathbf{C}\), where \(B=B^{\prime}\), and use the results obtained to prove the identity \(\sin a \cos b=\frac{1}{2} \sin (a+b)+\frac{1}{2} \sin (a-b)\)

The vectors P and Q are two adjacent sides of a parallelogram. Determine the area of the parallelogram when (a) P = -7i + 3j - 3k and Q = 2i + 2j + 5k, (b) P = 6i - 5j - 2k and Q = -2i + 5j - k.

A plane contains the vectors A and B. Determine the unit vector normal to the plane when A and B are equal to, respectively, (a) i + 2j - 5k and 4i - 7j - 5k, (b) 3i - 3j + 2k and -2i + 6j - 4k.

A line passes through the points (20 m, 16 m) and (-1 m, -4 m). Determine the perpendicular distance d from the line to the origin O of the system of coordinates.

Determine the moment about the origin O of the force F = 4i - 3j + 5k that acts at a point A. Assume that the position vector of A is (a) r = 2i + 3j - 4k, (b) r = -8i + 6j - 10k, (c) r = 8i - 6j + 5k.

Determine the moment about the origin O of the force F = 2i + 3j - 4k that acts at a point A. Assume that the position vector of A is (a) r = 3i - 6j + 5k, (b) r = i - 4j - 2k, (c) r = 4i + 6j - 8k.

The wire AE is stretched between the corners A and E of a bent plate. Knowing that the tension in the wire is 435 N, determine the moment about O of the force exerted by the wire (a) on corner A, (b) on corner E.

A small boat hangs from two davits, one of which is shown in the figure. The tension in line ABAD is 82 lb. Determine the moment about C of the resultant force \(\mathbf{R}_{{A}}\) exerted on the davit at A.

A 6-ft-long fishing rod AB is securely anchored in the sand of a beach. After a fish takes the bait, the resulting force in the line is 6 lb. Determine the moment about A of the force exerted by the line at B.

A precast concrete wall section is temporarily held by two cables as shown. Knowing that the tension in cable BD is 900 N, determine the moment about point O of the force exerted by the cable at B.

A 200-N force is applied as shown to the bracket ABC. Determine the moment of the force about A.

The 6-m boom AB has a fixed end A. A steel cable is stretched from the free end B of the boom to a point C located on the vertical wall. If the tension in the cable is 2.5 kN, determine the moment about A of the force exerted by the cable at B.

In Prob. 3.21, determine the perpendicular distance from point O to wire AE.

In Prob. 3.21, determine the perpendicular distance from point B to wire AE.

In Prob. 3.22, determine the perpendicular distance from point C to portion AD of the line ABAD.

In Prob. 3.23, determine the perpendicular distance from point A to a line drawn through points B and C.

In Prob. 3.23, determine the perpendicular distance from point D to a line drawn through points B and C.

In Prob. 3.24, determine the perpendicular distance from point O to cable BD.

In Prob. 3.24, determine the perpendicular distance from point C to cable BD.

Determine the value of a that minimizes the perpendicular distance from point C to a section of pipeline that passes through points A and B.

Given the vectors P = 3i - j + 2k, Q = 4i + 5j - 3k, and S = -2i + 3j - k, compute the scalar products \(\mathbf {P} \cdot \mathbf {Q}\), \(\mathbf {P} \cdot \mathbf {S}\), and \(\mathbf {Q} \cdot \mathbf {S}\).

Form the scalar product \(\mathbf {B} \cdot \mathbf {C}\) and use the result obtained to prove the identity cos (a - b) = cos a cos b + sin a sin b

Consider the volleyball net shown. Determine the angle formed by guy wires AB and AC.

Consider the volleyball net shown. Determine the angle formed by guy wires AC and AD.

Three cables are used to support a container as shown. Determine the angle formed by cables AB and AD.

Three cables are used to support a container as shown. Determine the angle formed by cables AC and AD.

The 20-in. tube AB can slide along a horizontal rod. The ends A and B of the tube are connected by elastic cords to the fixed point C. For the position corresponding to x = 11 in., determine the angle formed by the two cords, (a) using Eq. (3.32), (b) applying the law of cosines to triangle ABC.

Solve Prob. 3.41 for the position corresponding to x = 4 in.

Ropes AB and BC are two of the ropes used to support a tent. The two ropes are attached to a stake at B. If the tension in rope AB is 540 N, determine (a) the angle between rope AB and the stake, (b) the projection on the stake of the force exerted by rope AB at point B.

Ropes AB and BC are two of the ropes used to support a tent. The two ropes are attached to a stake at B. If the tension in rope BC is 490 N, determine (a) the angle between rope BC and the stake, (b) the projection on the stake of the force exerted by rope BC at point B.

Given the vectors P = 4i - 2j + 3k, Q = 2i + 4j - 5k, and S = \(S_{x} \mathbf{i}-\mathbf{j}+2 \mathbf{k}\), determine the value of \(S_{x}\) for which the three vectors are coplanar.

Determine the volume of the parallelepiped of Fig. 3.25 when (a) P = 4i - 3j + 2k, Q = -2i - 5j + k, and S = 7i + j - k, (b) P = 5i - j + 6k, Q = 2i + 3j + k, and S = -3i - 2j + 4k

Knowing that the tension in cable AB is 570 N, determine the moment about each of the coordinate axes of the force exerted on the plate at B.

Knowing that the tension in cable AC is 1065 N, determine the moment about each of the coordinate axes of the force exerted on the plate at C.

A small boat hangs from two davits, one of which is shown in the figure. It is known that the moment about the z axis of the resultant force \(\mathbf{R}_{A}\) exerted on the davit at A must not exceed \(279\mathrm{\ lb}\cdot\mathrm{ft}\) in absolute value. Determine the largest allowable tension in line ABAD when x = 6 ft.

For the davit of Prob. 3.49, determine the largest allowable distance x when the tension in line ABAD is 60 lb.

A farmer uses cables and winch pullers B and E to plumb one side of a small barn. If it is known that the sum of the moments about the x axis of the forces exerted by the cables on the barn at points A and D is equal to \(4728\ \mathrm{lb}\cdot\mathrm{ft}\), determine the magnitude of \(\mathbf{T}_{D E}\) when \(T_{AB}=255\mathrm{\ lb}\).

Solve Prob. 3.51 when the tension in cable AB is 306 lb.

A single force P acts at C in a direction perpendicular to the handle BC of the crank shown. Knowing that \(M_x=+20\mathrm{\ N}\cdot\mathrm{m}\) and \(M_y=-8.75\mathrm{\ N}\cdot\mathrm{m}\), and \(M_z=-30\mathrm{\ N}\cdot\mathrm{m}\), determine the magnitude of P and the values of f and u.

A single force P acts at C in a direction perpendicular to the handle BC of the crank shown. Determine the moment \(M_x\) of P about the x axis when \(u=65^{\circ}\), knowing that \(M_y=-15\mathrm{\ N}\cdot\mathrm{m}\) and \(M_z=-36\mathrm{\ N}\cdot\mathrm{m}\).

The triangular plate ABC is supported by ball-and-socket joints at B and D and is held in the position shown by cables AE and CF. If the force exerted by cable AE at A is 55 N, determine the moment of that force about the line joining points D and B.

The triangular plate ABC is supported by ball-and-socket joints at B and D and is held in the position shown by cables AE and CF. If the force exerted by cable CF at C is 33 N, determine the moment of that force about the line joining points D and B.

The 23-in. vertical rod CD is welded to the midpoint C of the 50-in. rod AB. Determine the moment about AB of the 235-lb force P.

The 23-in. vertical rod CD is welded to the midpoint C of the 50-in. rod AB. Determine the moment about AB of the 174-lb force Q.

The frame ACD is hinged at A and D and is supported by a cable that passes through a ring at B and is attached to hooks at G and H. Knowing that the tension in the cable is 450 N, determine the moment about the diagonal AD of the force exerted on the frame by portion BH of the cable.

In Prob. 3.59, determine the moment about the diagonal AD of the force exerted on the frame by portion BG of the cable.

A regular tetrahedron has six edges of length a. A force P is directed as shown along edge BC. Determine the moment of P about edge OA.

A regular tetrahedron has six edges of length a. (a) Show that two opposite edges, such as OA and BC, are perpendicular to each other. (b) Use this property and the result obtained in Prob. 3.61 to determine the perpendicular distance between edges OA and BC.

Two forces \(\mathbf{F}_{1}\) and \(\mathbf{F}_{2}\) in space have the same magnitude F. Prove that the moment of \(\mathbf{F}_{1}\) about the line of action of \(\mathbf{F}_{2}\) is equal to the moment of \(\mathbf{F}_{2}\) about the line of action of \(\mathbf{F}_{1}\).

In Prob. 3.55, determine the perpendicular distance between cable AE and the line joining points D and B.

In Prob. 3.56, determine the perpendicular distance between cable CF and the line joining points D and B.

In Prob. 3.57, determine the perpendicular distance between rod AB and the line of action of P.

In Prob. 3.58, determine the perpendicular distance between rod AB and the line of action of Q.

In Prob. 3.59, determine the perpendicular distance between portion BH of the cable and the diagonal AD.

In Prob. 3.60, determine the perpendicular distance between portion BG of the cable and the diagonal AD.

A plate in the shape of a parallelogram is acted upon by two couples. Determine (a) the moment of the couple formed by the two 21-lb forces, (b) the perpendicular distance between the 12-lb forces if the resultant of the two couples is zero, (c) the value of a if the resultant couple is \(72\mathrm{\ lb}\cdot\mathrm{in.}\) clockwise and d is 42 in.

Four 1-in.-diameter pegs are attached to a board as shown. Two strings are passed around the pegs and pulled with the forces indicated. (a) Determine the resultant couple acting on the board. (b) If only one string is used, around which pegs should it pass and in what directions should it be pulled to create the same couple with the minimum tension in the string? (c) What is the value of that minimum tension?

Four pegs of the same diameter are attached to a board as shown. Two strings are passed around the pegs and pulled with the forces indicated. Determine the diameter of the pegs knowing that the resultant couple applied to the board is \(485\ \mathrm{lb}\cdot\mathrm{in}.\) counterclockwise.

A piece of plywood in which several holes are being drilled successively has been secured to a workbench by means of two nails. Knowing that the drill exerts a \(12-\mathrm{N} \cdot \mathrm{m}\) couple on the piece of plywood, determine the magnitude of the resulting forces applied to the nails if they are located (a) at A and B, (b) at B and C, (c) at A and C.

Two parallel 40-N forces are applied to a lever as shown. Determine the moment of the couple formed by the two forces (a) by resolving each force into horizontal and vertical components and adding the moments of the two resulting couples, (b) by using the perpendicular distance between the two forces, (c) by summing the moments of the two forces about point A.

The two shafts of a speed-reducer unit are subjected to couples of magnitude \(M_1=15\mathrm{\ lb}\cdot\mathrm{ft}\) and \(M_2=3\mathrm{\ lb}\cdot\mathrm{ft}\), respectively. Replace the two couples with a single equivalent couple, specifying its magnitude and the direction of its axis.

Replace the two couples shown with a single equivalent couple, specifying its magnitude and the direction of its axis.

Solve Prob. 3.76, assuming that two 10-N vertical forces have been added, one acting upward at C and the other downward at B.

If P = 0, replace the two remaining couples with a single equivalent couple, specifying its magnitude and the direction of its axis.

If P = 20 lb, replace the three remaining couples with a single equivalent couple, specifying its magnitude and the direction of its axis.

In a manufacturing operation, three holes are drilled simultaneously in a workpiece. If the holes are perpendicular to the surfaces of the workpiece, replace the couples applied to the drills with a single equivalent couple, specifying its magnitude and the direction of its axis.

A 260-lb force is applied at A to the rolled-steel section shown. Replace that force with an equivalent force-couple system at the center C of the section.

A 30-lb vertical force P is applied at A to the bracket shown, which is held by screws at B and C. (a) Replace P with an equivalent force-couple system at B. (b) Find the two horizontal forces at B and C that are equivalent to the couple obtained in part a.

The force P has a magnitude of 250 N and is applied at the end C of a 500-mm rod AC attached to a bracket at A and B. Assuming \(a=30^{\circ}\) and \(b=60^{\circ}\), replace P with (a) an equivalent force-couple system at B, (b) an equivalent system formed by two parallel forces applied at A and B.

Solve Prob. 3.83, assuming \(a=b=25^{\circ}\).

The 80-N horizontal force P acts on a bell crank as shown. (a) Replace P with an equivalent force-couple system at B. (b) Find the two vertical forces at C and D that are equivalent to the couple found in part a.

A dirigible is tethered by a cable attached to its cabin at B. If the tension in the cable is 1040 N, replace the force exerted by the cable at B with an equivalent system formed by two parallel forces applied at A and C.

Three control rods attached to a lever ABC exert on it the forces shown. (a) Replace the three forces with an equivalent force-couple system at B. (b) Determine the single force that is equivalent to the force-couple system obtained in part a, and specify its point of application on the lever.

A hexagonal plate is acted upon by the force P and the couple shown. Determine the magnitude and the direction of the smallest force P for which this system can be replaced with a single force at E.

A force and couple act as shown on a square plate of side a = 25 in. Knowing that P = 60 lb, Q = 40 lb, and \(a=50^{\circ}\), replace the given force and couple with a single force applied at a point located (a) on line AB, (b) on line AC. In each case determine the distance from A to the point of application of the force.

The force and couple shown are to be replaced by an equivalent single force. Knowing that P = 2Q, determine the required value of a if the line of action of the single equivalent force is to pass through (a) point A, (b) point C.

The shearing forces exerted on the cross section of a steel channel can be represented by a 900-N vertical force and two 250-N horizontal forces as shown. Replace this force and couple with a single force F applied at point C, and determine the distance x from C to line BD. (Point C is defined as the shear center of the section.)

A force and a couple are applied as shown to the end of a cantilever beam. (a) Replace this system with a single force F applied at point C, and determine the distance d from C to a line drawn through points D and E. (b) Solve part a if the directions of the two 360-N forces are reversed.

An antenna is guyed by three cables as shown. Knowing that the tension in cable AB is 288 lb, replace the force exerted at A by cable AB with an equivalent force-couple system at the center O of the base of the antenna.

An antenna is guyed by three cables as shown. Knowing that the tension in cable AD is 270 lb, replace the force exerted at A by cable AD with an equivalent force-couple system at the center O of the base of the antenna.

A 110-N force acting in a vertical plane parallel to the yz plane is applied to the 220-mm-long horizontal handle AB of a socket wrench. Replace the force with an equivalent force-couple system at the origin O of the coordinate system.

An eccentric, compressive 1220-N force P is applied to the end of a cantilever beam. Replace P with an equivalent force-couple system at G.

To keep a door closed, a wooden stick is wedged between the floor and the doorknob. The stick exerts at B a 175-N force directed along line AB. Replace that force with an equivalent force-couple system at C.

A 46-lb force F and a \(2120-\mathrm{lb} \cdot \mathrm{in} .\) couple M are applied to corner A of the block shown. Replace the given force-couple system with an equivalent force-couple system at corner H.

A 77-N force \(\mathbf{F}_{1}\) and a \(31-\mathrm{N} \cdot \mathrm{m}\) couple \(\mathbf{M}_{1}\) are applied to corner E of the bent plate shown. If \(\mathbf{F}_{1}\) and \(\mathbf{M}_{1}\) are to be replaced with an equivalent force-couple system (\(\mathbf{F}_{2}\), \(\mathbf{M}_{2}\)) at corner B and if \(\left(M_{2}\right)_{z}=0\), determine (a) the distance d, (b) \(\mathbf{F}_{2}\) and \(\mathbf{M}_{2}\).

A 2.6-kip force is applied at point D of the cast-iron post shown. Replace that force with an equivalent force-couple system at the center A of the base section.

A 3-m-long beam is subjected to a variety of loadings. (a) Replace each loading with an equivalent force-couple system at end A of the beam. (b) Which of the loadings are equivalent?

A 3-m-long beam is loaded as shown. Determine the loading of Prob. 3.101 that is equivalent to this loading.

Determine the single equivalent force and the distance from point A to its line of action for the beam and loading of (a) Prob. 3.101a, (b) Prob. 3.101b, (c) Prob. 3.102.

Five separate force-couple systems act at the corners of a piece of sheet metal, which has been bent into the shape shown. Determine which of these systems is equivalent to a force F = (10 lb)i and a couple of moment \(\mathbf{M}=(15\ \mathrm{lb}\cdot\mathrm{ft})\mathbf{j}+(15\mathrm{\ lb}\cdot\mathrm{ft})\mathbf{k}\) located at the origin.

Three horizontal forces are applied as shown to a vertical cast-iron arm. Determine the resultant of the forces and the distance from the ground to its line of action when (a) P = 200 N, (b) P = 2400 N, (c) P = 1000 N.

Three stage lights are mounted on a pipe as shown. The lights at A and B each weigh 4.1 lb, while the one at C weighs 3.5 lb. (a) If d = 25 in., determine the distance from D to the line of action of the resultant of the weights of the three lights. (b) Determine the value of d so that the resultant of the weights passes through the midpoint of the pipe.

The weights of two children sitting at ends A and B of a seesaw are 84 lb and 64 lb, respectively. Where should a third child sit so that the resultant of the weights of the three children will pass through C if she weighs (a) 60 lb, (b) 52 lb?

A couple of magnitude \(M=54\mathrm{\ lb}\cdot\mathrm{in.}\) and the three forces shown are applied to an angle bracket. (a) Find the resultant of this system of forces. (b) Locate the points where the line of action of the resultant intersects line AB and line BC.

A couple M and the three forces shown are applied to an angle bracket. Find the moment of the couple if the line of action of the resultant of the force system is to pass through (a) point A, (b) point B, (c) point C.

A 32-lb motor is mounted on the floor. Find the resultant of the weight and the forces exerted on the belt, and determine where the line of action of the resultant intersects the floor.

A machine component is subjected to the forces and couples shown. The component is to be held in place by a single rivet that can resist a force but not a couple. For P = 0, determine the location of the rivet hole if it is to be located (a) on line FG, (b) on line GH.

Solve Prob. 3.111, assuming that P = 60 N.

A truss supports the loading shown. Determine the equivalent force acting on the truss and the point of intersection of its line of action with a line drawn through points A and G.

Four ropes are attached to a crate and exert the forces shown. If the forces are to be replaced with a single equivalent force applied at a point on line AB, determine (a) the equivalent force and the distance from A to the point of application of the force when \(a=30^{\circ}\), (b) the value of a so that the single equivalent force is applied at point B.

Solve Prob. 3.114, assuming that the 90-lb force is removed.

Four forces act on a \(700 \times 375-\mathrm{mm}\) plate as shown. (a) Find the resultant of these forces. (b) Locate the two points where the line of action of the resultant intersects the edge of the plate.

Solve Prob. 3.116, assuming that the 760-N force is directed to the right.

As follower AB rolls along the surface of member C, it exerts a constant force F perpendicular to the surface. (a) Replace F with an equivalent force-couple system at the point D obtained by drawing the perpendicular from the point of contact to the x axis. (b) For a = 1 m and b = 2 m, determine the value of x for which the moment of the equivalent force-couple system at D is maximum.

As plastic bushings are inserted into a 60-mm-diameter cylindrical sheet metal enclosure, the insertion tools exert the forces shown on the enclosure. Each of the forces is parallel to one of the coordinate axes. Replace these forces with an equivalent force-couple system at C.

Two 150-mm-diameter pulleys are mounted on line shaft AD. The belts at B and C lie in vertical planes parallel to the yz plane. Replace the belt forces shown with an equivalent force-couple system at A.

Four forces are applied to the machine component ABDE as shown. Replace these forces with an equivalent force-couple system at A.

While using a pencil sharpener, a student applies the forces and couple shown. (a) Determine the forces exerted at B and C knowing that these forces and the couple are equivalent to a force-couple system at A consisting of the force \(\mathbf{R}=(2.6\ \mathrm{lb})\mathbf{i}+R_{y}\mathbf{j}-(0.7 \ \mathrm{lb})\mathbf{k}\) and the couple \(\mathbf{M}_A^R=M_x\mathbf{i}+(1.0\mathrm{\ lb}\cdot\mathrm{ft})\mathbf{j}-(0.72\ \mathrm{lb}\cdot\mathrm{ft})\mathbf{k}\). (b) Find the corresponding values of \(R_y\) and \(M_x\) .

A blade held in a brace is used to tighten a screw at A. (a) Determine the forces exerted at B and C, knowing that these forces are equivalent to a force-couple system at A consisting of \(\mathbf{R}=-(30\mathrm{\ N})\mathbf{i}+R_{y} \mathbf{j}+R_{z} \mathbf{k}\) and \(\mathbf{M}_A^R=-(12\mathrm{\ N}\cdot\mathrm{m})\mathbf{i}\). (b) Find the corresponding values of \(R_y\) and \(R_z\). (c) What is the orientation of the slot in the head of the screw for which the blade is least likely to slip when the brace is in the position shown?

In order to unscrew the tapped faucet A, a plumber uses two pipe wrenches as shown. By exerting a 40-lb force on each wrench, at a distance of 10 in. from the axis of the pipe and in a direction perpendicular to the pipe and to the wrench, he prevents the pipe from rotating, and thus avoids loosening or further tightening the joint between the pipe and the tapped elbow C. Determine (a) the angle u that the wrench at A should form with the vertical if elbow C is not to rotate about the vertical, (b) the force-couple system at C equivalent to the two 40-lb forces when this condition is satisfied.

Assuming \(\mathrm{u}=60^{\circ}\) in Prob. 3.124, replace the two 40-lb forces with an equivalent force-couple system at D and determine whether the plumber’s action tends to tighten or loosen the joint between (a) pipe CD and elbow D, (b) elbow D and pipe DE. Assume all threads to be right-handed.

As an adjustable brace BC is used to bring a wall into plumb, the force-couple system shown is exerted on the wall. Replace this force-couple system with an equivalent force-couple system at A if R = 21.2 lb and \(M=13.25\mathrm{\ lb}\cdot\mathrm{ft}\).

Three children are standing on a \(5 \times 5-\mathrm{m}\) raft. If the weights of the children at points A, B, and C are 375 N, 260 N, and 400 N, respectively, determine the magnitude and the point of application of the resultant of the three weights.

Three children are standing on a \(5 \times 5-\mathrm{m}\) raft. The weights of the children at points A, B, and C are 375 N, 260 N, and 400 N, respectively. If a fourth child of weight 425 N climbs onto the raft, determine where she should stand if the other children remain in the positions shown and the line of action of the resultant of the four weights is to pass through the center of the raft.

Four signs are mounted on a frame spanning a highway, and the magnitudes of the horizontal wind forces acting on the signs are as shown. Determine the magnitude and the point of application of the resultant of the four wind forces when a = 1 ft and b = 12 ft.

Four signs are mounted on a frame spanning a highway, and the magnitudes of the horizontal wind forces acting on the signs are as shown. Determine a and b so that the point of application of the resultant of the four forces is at G.

A group of students loads a \(2 \times 3.3-\mathrm{m}\) flatbed trailer with two \(0.66 \times 0.66 \times 0.66-\mathrm{m}\) boxes and one \(0.66 \times 0.66 \times 1.2-\mathrm{m}\) box. Each of the boxes at the rear of the trailer is positioned so that it is aligned with both the back and a side of the trailer. Determine the smallest load the students should place in a second \(0.66 \times 0.66 \times 1.2-\mathrm{m}\) box and where on the trailer they should secure it, without any part of the box overhanging the sides of the trailer, if each box is uniformly loaded and the line of action of the resultant of the weights of the four boxes is to pass through the point of intersection of the centerlines of the trailer and the axle. (Hint: Keep in mind that the box may be placed either on its side or on its end.)

Solve Prob. 3.131 if the students want to place as much weight as possible in the fourth box and at least one side of the box must coincide with a side of the trailer.

A piece of sheet metal is bent into the shape shown and is acted upon by three forces. If the forces have the same magnitude P, replace them with an equivalent wrench and determine (a) the magnitude and the direction of the resultant force R, (b) the pitch of the wrench, (c) the axis of the wrench.

Three forces of the same magnitude P act on a cube of side a as shown. Replace the three forces with an equivalent wrench and determine (a) the magnitude and direction of the resultant force R, (b) the pitch of the wrench, (c) the axis of the wrench.

The forces and couples shown are applied to two screws as a piece of sheet metal is fastened to a block of wood. Reduce the forces and the couples to an equivalent wrench and determine (a) the resultant force R, (b) the pitch of the wrench, (c) the point where the axis of the wrench intersects the xz plane.

The forces and couples shown are applied to two screws as a piece of sheet metal is fastened to a block of wood. Reduce the forces and the couples to an equivalent wrench and determine (a) the resultant force R, (b) the pitch of the wrench, (c) the point where the axis of the wrench intersects the xz plane.

Two bolts at A and B are tightened by applying the forces and couples shown. Replace the two wrenches with a single equivalent wrench and determine (a) the resultant R, (b) the pitch of the single equivalent wrench, (c) the point where the axis of the wrench intersects the xz plane.

Two bolts at A and B are tightened by applying the forces and couples shown. Replace the two wrenches with a single equivalent wrench and determine (a) the resultant R, (b) the pitch of the single equivalent wrench, (c) the point where the axis of the wrench intersects the xz plane.

A flagpole is guyed by three cables. If the tensions in the cables have the same magnitude P, replace the forces exerted on the pole with an equivalent wrench and determine (a) the resultant force R, (b) the pitch of the wrench, (c) the point where the axis of the wrench intersects the xz plane.

Two ropes attached at A and B are used to move the trunk of a fallen tree. Replace the forces exerted by the ropes with an equivalent wrench and determine (a) the resultant force R, (b) the pitch of the wrench, (c) the point where the axis of the wrench intersects the yz plane.

Determine whether the force-and-couple system shown can be reduced to a single equivalent force R. If it can, determine R and the point where the line of action of R intersects the yz plane. If it cannot be so reduced, replace the given system with an equivalent wrench and determine its resultant, its pitch, and the point where its axis intersects the yz plane.

Determine whether the force-and-couple system shown can be reduced to a single equivalent force R. If it can, determine R and the point where the line of action of R intersects the yz plane. If it cannot be so reduced, replace the given system with an equivalent wrench and determine its resultant, its pitch, and the point where its axis intersects the yz plane.

Replace the wrench shown with an equivalent system consisting of two forces perpendicular to the y axis and applied respectively at A and B.

Show that, in general, a wrench can be replaced with two forces chosen in such a way that one force passes through a given point while the other force lies in a given plane.

Show that a wrench can be replaced with two perpendicular forces, one of which is applied at a given point.

Show that a wrench can be replaced with two forces, one of which has a prescribed line of action.

A 300-N force is applied at A as shown. Determine (a) the moment of the 300-N force about D, (b) the smallest force applied at B that creates the same moment about D.

The tailgate of a car is supported by the hydraulic lift BC. If the lift exerts a 125-lb force directed along its centerline on the ball and socket at B, determine the moment of the force about A.

The ramp ABCD is supported by cables at corners C and D. The tension in each of the cables is 810 N. Determine the moment about A of the force exerted by (a) the cable at D, (b) the cable at C.

Section AB of a pipeline lies in the yz plane and forms an angle of \(37^{\circ}\) with the z axis. Branch lines CD and EF join AB as shown. Determine the angle formed by pipes AB and CD.

To lift a heavy crate, a man uses a block and tackle attached to the bottom of an I-beam at hook B. Knowing that the moments about the y and the z axes of the force exerted at B by portion AB of the rope are, respectively, \(120\mathrm{\ N}\cdot\mathrm{m}\) and \(-460\mathrm{\ N}\cdot\mathrm{m}\), determine the distance a.

To loosen a frozen valve, a force F of magnitude 70 lb is applied to the handle of the valve. Knowing that \(\mathrm{u}=25^{\circ}\), \(M_x=-61\ \mathrm{lb}\cdot\mathrm{ft}\), and \(M_z=-43\ \mathrm{lb}\cdot\mathrm{ft}\), determine f and d.

The tension in the cable attached to the end C of an adjustable boom ABC is 560 lb. Replace the force exerted by the cable at C with an equivalent force-couple system (a) at A, (b) at B.

While tapping a hole, a machinist applies the horizontal forces shown to the handle of the tap wrench. Show that these forces are equivalent to a single force, and specify, if possible, the point of application of the single force on the handle.

Replace the 150-N force with an equivalent force-couple system at A.

A beam supports three loads of given magnitude and a fourth load whose magnitude is a function of position. If b = 1.5 m and the loads are to be replaced with a single equivalent force, determine (a) the value of a so that the distance from support A to the line of action of the equivalent force is maximum, (b) the magnitude of the equivalent force and its point of application on the beam.

A mechanic uses a crowfoot wrench to loosen a bolt at C. The mechanic holds the socket wrench handle at points A and B and applies forces at these points. Knowing that these forces are equivalent to a force-couple system at C consisting of the force \(\mathbf{C}=-(8\ \mathrm{lb})\mathbf{i}+(4\mathrm{\ lb})\mathbf{k}\) and the couple \(\mathbf{M}_C=(360\mathrm{\ lb}\cdot\mathrm{in}.)\mathbf{i}\), determine the forces applied at A and at B when \(A_z=2\mathrm{\ lb}\).

A concrete foundation mat in the shape of a regular hexagon of side 12 ft supports four column loads as shown. Determine the magnitudes of the additional loads that must be applied at B and F if the resultant of all six loads is to pass through the center of the mat.