Two parallel forces P and Q are applied at the ends of a beam AB of length L. Find the

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 \(\alpha = 25^\circ\), 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 the moment of the force about B is \(120 \ \mathrm{lb} \cdot \mathrm{in}\). clockwise, determine the value of \(\alpha\).

For the brake pedal shown, determine the magnitude and direction of the smallest force P that has a \(104-\mathrm{N} \cdot \mathrm{m}\) clockwise moment about B.

A force P is applied to the brake pedal at A. Knowing that P = 450 N and \(\alpha = 30^\circ\), determine the moment of P about B.

A 450-N force is applied at A as shown. Determine (a) the moment of the 450-N force about D, (b) the smallest force applied at B that creates the same moment about D.

A 450-N force is applied at A as shown. Determine (a) the moment of the 450-N force about D, (b) the magnitude and sense of the horizontal force applied at C that creates the same moment about D, (c) the smallest force applied at C that creates the same moment about D.

Compute the moment of the 100-lb force about A, (a) by using the definition of the moment of a force, (b) by resolving the force into horizontal and vertical components, (c) by resolving the force into components along AB and in the direction perpendicular to AB.

Determine the moment of the 100-lb force about C.

It is known that the connecting rod AB exerts on the crank BC a 2.5-kN force directed down to the left along the centerline AB. Determine the moment of that force about C.

It is known that the connecting rod AB exerts on the crank BC a 2.5-kN force directed down to the left along the centerline AB. Determine the moment of that force about C.

Rod AB is held in place by the cord AC. Knowing that the tension in the cord is 300 lb and that c = 18 in., determine the moment about B of the force exerted by the cord at point A by resolving that force into horizontal and vertical components applied (a) at point A, (b) at point C.

Rod AB is held in place by the cord AC. Knowing that c 5 42 in. and that the moment about B of the force exerted by the cord at point A is \(700 \ \mathrm{lb} \cdot \mathrm{ft}\), determine the tension in the cord.

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

Determine the moment about the origin of coordinates O of the force F = -i + 3j + 5k that acts at a point A. Assume that the position of A is (a) r = 2i - 4j + k, (b) r = 4i + 6j + 10k, (c) r = -3i + 9j + 15k.

The line of action of the force P of magnitude 420 lb passes through the two points A and B as shown. Compute the moment of P about O using the position vector (a) of point A, (b) of point B.

A force P of magnitude 200 N acts along the diagonal BC of the bent plate shown. Determine the moment of P about point E.

Knowing that the tension in cable AB is 1800 lb, determine the moment of the force exerted on the plate at A about (a) the origin of coordinates O, (b) corner D.

Knowing that the tension in cable BC is 900 lb, determine the moment of the force exerted on the plate at C about (a) the origin of coordinates O, (b) corner D.

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

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.

In Prob. 3.15, determine the perpendicular distance from the line of action of P to the origin O.

In Prob. 3.16, determine the perpendicular distance from the line of action of P to point E.

In Prob. 3.20, determine the perpendicular distance from the point C to the portion AD of line ABAD.

In Sample Prob. 3.4, determine the perpendicular distance from point A to wire CD.

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

Form the scalar product \(\mathbf{P}_1 \cdot \mathbf{P}_2\), and use the result obtained to prove the identity \(\cos \left(\theta_1-\theta_2\right)=\cos \theta_1 \cos \theta_2+\sin \theta_1 \sin \theta_2\).

Knowing that the tension in cable BC is 1400 N, determine (a) the angle between cable BC and the boom AB, (b) the projection on AB of the force exerted by cable BC at point B.

Knowing that the tension in cable BD is 900 N, determine (a) the angle between cable BD and the boom AB, (b) the projection on AB of the force exerted by cable BD at point B.

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 500-mm 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 = 275 mm, determine the angle formed by the two cords, (a) using Eq. (3.32), (b) applying the law of cosines to triangle ABC.

Solve Prob. 3.31 for the position corresponding to x = 100 mm.

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

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

The jib crane is oriented so that the boom DA is parallel to the x axis. At the instant shown, the tension in cable AB is 13 kN. Determine the moment about each of the coordinate axes of the force exerted on A by the cable AB.

The jib crane is oriented so that the boom DA is parallel to the x axis. Determine the maximum permissible tension in the cable AB if the absolute values of the moments about the coordinate axes of the force exerted on A must be as follows: \(\left|M_x\right| \leq 10 \mathrm{kN} \cdot \mathrm{m}\), \(\left|M_y\right| \leq 6 \mathrm{kN} \cdot \mathrm{m}\), and \(\left|M_z\right| \leq 16 \mathrm{kN} \cdot \mathrm{m}\).

The primary purpose of the crank shown is to produce a moment about the x axis. Show that a single force acting at A and having moment \(M_x\) different from zero about the x axis must also have a moment different from zero about at least one of the other coordinate axes.

A single force F of unknown magnitude and direction acts at point A of the crank shown. Determine the moment \(M_x\) of F about the x axis knowing that \(M_y=+180 \mathrm{lb} \cdot\) in. and \(M_z=-320 \mathrm{lb} \cdot\) in.

The rectangular platform is hinged at A and B and supported by a cable that passes over a frictionless hook at E. Knowing that the tension in the cable is 1349 N, determine the moment about each of the coordinate axes of the force exerted by the cable at C.

For the platform of Prob. 3.39, determine the moment about each of the coordinate axes of the force exerted by the cable at D.

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 \(279 \ \mathrm{lb} \cdot \mathrm{f}\)t in absolute value. Determine the largest allowable tension in the line ABAD when x = 6 ft.

For the davit of Prob. 3.41, determine the largest allowable distance x when the tension in the line ABAD is 60 lb.

A force P of magnitude 25 lb acts on a bent rod as shown. Determine the moment of P about (a) a line joining points C and F, (b) a line joining points O and C.

A force P of magnitude 25 lb acts on a bent rod as shown. Determine the moment of P about (a) a line joining points A and C, (b) a line joining points A and D.

Two rods are welded together to form a T-shaped lever that is acted upon by a 650-N force as shown. Determine the moment of the force about rod AB.

The rectangular plate ABCD is held by hinges along its edge AD and by the wire BE. Knowing that the tension in the wire is 546 N, determine the moment about AD of the force exerted by the wire at point 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.

A couple formed by two 975-N forces is applied to the pulley assembly shown. Determine an equivalent couple that is formed by (a) vertical forces acting at A and C, (b) the smallest possible forces acting at B and D, (c) the smallest possible forces that can be attached to the assembly.

Four 1-in.-diameter pegs are attached to a board as shown. Two strings are passed around the pegs and pulled with forces of magnitude P = 20 lb and Q = 35 lb. Determine the resultant couple acting on the board.

Two 80-N forces are applied as shown to the corners B and D of a rectangular plate. (a) Determine the moment of the couple formed by the two forces by resolving each force into horizontal and vertical components and adding the moments of the two resulting couples. (b) Use the result obtained to determine the perpendicular distance between lines BE and DF.

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.

Four \(1 \frac{1}{2}\)-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 \(1132.5 \mathrm{lb} \cdot \mathrm{in}.\) counterclockwise.

The axles and drive shaft of a rear-wheel drive automobile are acted upon by the three couples shown. Replace these three couples by a single equivalent couple.

Two shafts for a speed-reducer unit are subjected to couples of magnitude \(M_1 = 12 \ \mathrm{lb} \cdot \mathrm{ft}\) and \(M_2 = 5 \ \mathrm{lb} \cdot \mathrm{ft}\). Replace the two couples by a single equivalent couple, specifying its magnitude and the direction of its axis.

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

Solve Prob. 3.57 assuming that two 10-N vertical forces have been added, one acting upward at C and the other downward at B.

Shafts A and B connect the gear box to the wheel assemblies of a tractor, and shaft C connects it to the engine. Shafts A and B lie in the vertical yz plane, while shaft C is directed along the x axis. Replace the couples applied to the shafts by a single equivalent couple, specifying its magnitude and the direction of its axis.

\(\mathbf{M}_1\) and \(\mathbf{M}_2\) represent couples that are contained in the planes ABC and ACD, respectively. Assuming that \(M_1 = M_2 = M\), determine a single couple equivalent to the two given couples.

A 60-lb vertical force P is applied at A to the bracket shown, which is held by screws at B and C. (a) Replace P by 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 and couple shown are to be replaced by an equivalent single force. Determine the required value of \(\alpha\) so that the line of action of the single equivalent force will pass through point B.

Knowing that \(\alpha = 60^\circ\), replace the force and couple shown by a single force applied at a point located (a) on line AB, (b) on line CD. In each case determine the distance from the center O to the point of application of the force.

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

Force P has a magnitude of 300 N and is applied at A in a direction perpendicular to the handle \((\alpha = 0)\). Assuming \(\beta = 30^\circ\), replace force P by (a) an equivalent force-couple system at B, (b) an equivalent system formed by two parallel forces applied at B and C.

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 \(\alpha = 50^\circ\), replace the given force and couple by 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.

Replace the 250-kN force P by an equivalent force-couple system at G.

A 4-kip force is applied on the outside face of the flange of a steel channel. Determine the components of the force and couple at G that are equivalent to the 4-kip load.

The 12-ft boom AB has a fixed end A, and the tension in cable BC is 570 lb. Replace the force that the cable exerts at B by an equivalent force-couple system at A.

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

The jib crane shown is orientated so that its boom AD is parallel to the x axis and is used to move a heavy crate. Knowing that the tension in cable AB is 2.6 kips, replace the force exerted by the cable at A by an equivalent force-couple system at the center O of the base of the crane.

A 200-N force is applied as shown on the bracket ABC. Determine the components of the force and couple at A that are equivalent to this force.

A 12-ft beam is loaded in the various ways represented in the figure. Find two loadings that are equivalent.

A 12-ft beam is loaded as shown. Determine the loading of Prob. 3.73 that is equivalent to this loading.

By driving the truck shown over a scale, it was determined that the loads on the front and rear axles are, respectively, 18 kN and 12 kN when the truck is empty. Determine (a) the location of the center of gravity of the truck, (b) the weight and location of the center of gravity of the heaviest load that can be carried by the truck if the load on each axle is not to exceed 40 kN.

Four packages are transported at constant speed from A to B by the conveyor. At the instant shown, determine the resultant of the loading and the location of its line of action.

Determine the distance from point A to the line of action of the resultant of the three forces shown when (a) a = 1 m, (b) a = 1.5 m, (c) a = 2.5 m.

Two parallel forces P and Q are applied at the ends of a beam AB of length L. Find the distance x from A to the line of action of their resultant. Check the formula obtained by assuming L = 200 mm and (a) P = 50 N down, Q = 150 N down; (b) P = 50 N down, Q = 150 N up.

Three forces act as shown on a traffic-signal pole. Determine (a) the equivalent force-couple system at A, (b) the resultant of the system and the point of intersection of its line of action with the pole.

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.

The three forces shown and a couple of magnitude \(M = 80 \ \mathrm{lb} \cdot \mathrm{in.}\) 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 bracket is subjected to the system of forces and couples shown. Find the resultant of the system and the point of intersection of its line of action with (a) line AB, (b) line BC, (c) line CD.

The roof of a building frame is subjected to the wind loading shown. Determine (a) the equivalent force-couple system at D, (b) the resultant of the loading and its line of action.

Two cables exert forces of 90 kN each on a truss of weight W = 200 kN. Find the resultant force acting on the truss and the point of intersection of its line of action with line AB.

Two forces are applied to the vertical post as shown. Determine the force and couple at O equivalent to the two forces.

In order to move a 70.6-kg crate, two men push on it while two other men pull on it by means of ropes. The force exerted by man A is 600 N and that exerted by man B is 200 N; both forces are horizontal. Man C pulls with a force equal to 320 N and man D with a force equal to 480 N. Both cables form an angle of \(30^\circ\) with the vertical. Determine the resultant of all the forces acting on the crate.

The machine component is subject to the forces shown, each of which is parallel to one of the coordinate axes. Replace these forces by an equivalent force-couple system at A.

In drilling a hole in a wall, a man applies a vertical 30-lb force at B on the brace and bit, while pushing at C with a 10-lb force. The brace lies in the horizontal xz plane. (a) Determine the other components of the total force that should be exerted at C if the bit is not to be bent about the y and z axes (i.e., if the system of forces applied on the brace is to have zero moment about both the y and z axes). (b) Reduce the 30-lb force and the total force at C to an equivalent force and couple at A.

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, the plumber 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 \(\theta\) 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 \(\theta = 60^\circ\) in Prob. 3.89, replace the two 40-lb forces by 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 the threads to be right-handed.

A rectangular concrete foundation mat supports four column loads as shown. Determine the magnitude and point of application of the resultant of the four loads.

A concrete foundation mat in the shape of a regular hexagon of 10-ft sides supports four column loads as shown. Determine the magnitude and point of application of the resultant of the four loads.

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.

In Prob. 3.91, determine the magnitude and point of application of the smallest additional load that must be applied to the foundation mat if the resultant of the five loads is to pass through the center of the mat.

Four horizontal forces act on a vertical quarter-circular plate of radius 250 mm. Determine the magnitude and point of application of the resultant of the four forces if P = 40 N.

Determine the magnitude of the force P for which the resultant of the four forces acts on the rim of the plate.

A force P of magnitude 520 lb acts on the frame shown at point E. Determine the moment of P (a) about point D, (b) about a line joining points O and D.

A force P acts on the frame shown at point E. Knowing that the absolute value of the moment of P about a line joining points F and B is \(300 \ \mathrm{lb} \cdot \mathrm{ft}\), determine the magnitude of the force P.

A crane is oriented so that the end of the 25-m boom AO lies in the yz plane. At the instant shown the tension in cable AB is 4 kN. Determine the moment about each of the coordinate axes of the force exerted on A by cable AB.

The 25-m crane boom AO lies in the yz plane. Determine the maximum permissible tension in cable AB if the absolute value of the moments about the coordinate axes of the force exerted on A by cable AB must be as follows: \(\left|M_x\right| \leq 60 \mathrm{kN} \cdot \mathrm{m},\left|M_y\right| \leq 12 \mathrm{kN} \cdot \mathrm{m}\), and \(\left|M_z\right| \leq 8 \mathrm{kN} \cdot \mathrm{m}\).

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 \(\theta=65^{\circ}\) knowing that \(M_y=-15 \mathrm{~N} \cdot \mathrm{m}\) and \(M_z= -36 \mathrm{~N} \cdot \mathrm{m}\).

A multiple-drilling machine is used to drill simultaneously six holes in the steel plate shown. Each drill exerts a clockwise couple of magnitude \(40 \ \mathrm{lb} \cdot \mathrm{in.}\) on the plate. Determine an equivalent couple formed by the smallest possible forces acting (a) at A and C, (b) at A and D, (c) on the plate.

A 500-N force is applied to a bent plate as shown. Determine (a) an equivalent force-couple system at B, (b) an equivalent system formed by a vertical force at A and a force at B.

A 100-kN load is applied eccentrically to the column shown. Determine the components of the force and couple at G that are equivalent to the 100-kN load.

The speed-reducer unit shown weighs 75 lb, and its center of gravity is located on the y axis. Show that the weight of the unit and the two couples acting on it, of magnitude \(M_1 = 20 \ \mathrm{lb} \cdot \mathrm{ft}\) and \(M_2 = 4 \ \mathrm{lb} \cdot \mathrm{ft}\), respectively, can be replaced by a single equivalent force and determine (a) the magnitude and direction of that force, (b) the point where its line of action intersects the floor.

For the truss and loading shown, determine the resultant of the loads and the distance from point A to its line of action.

A force P of given magnitude P is applied to the edge of a semicircular plate of radius a as shown. (a) Replace P by an equivalent force-couple system at point D obtained by drawing the perpendicular from B to the x axis. (b) Determine the value of \(\theta\) for which the moment of the equivalent force-couple system at D is maximum.

A concrete foundation mat of 5-m radius supports four equally spaced columns, each of which is located 4 m from the center of the mat. Determine the magnitude and point of application of the smallest additional load that must be applied to the foundation mat if the resultant of the five loads is to pass through the center of the mat.