Solved: If the resultant couple of the three couples

If A, B, and D are given vectors, prove the distributive law for the vector cross product, i.e., \(A \times (B + D) = (A \times B) + (A \times D)\).

Prove the triple scalar product identity \(A \cdot (B \times C) = (A \times B) \cdot C\).

Given the three nonzero vectors A, B, and C, show that if \(A \cdot (B \times C) = 0\), the three vectors must lie in the same plane.

Determine the moment about point A of each of the three forces acting on the beam.

Determine the moment about point B of each of the three forces acting on the beam.

The crowbar is subjected to a vertical force of P = 25 lb at the grip, whereas it takes a force of F = 155 lb at the claw to pull the nail out. Find the moment of each force about point A and determine if P is sufficient to pull out the nail. The crowbar contacts the board at point A.

Determine the moment of each of the three forces about point A.

Determine the moment of each of the three forces about point B.

Determine the moment of each force about the bolt located at A. Take \(F_B = 40\ lb\), \(F_C = 50\ lb\).

If \(F_B = 30\ lb\) and \(F_C = 45\ lb\), determine the resultant moment about the bolt located at A.

The towline exerts a force of P = 6 kN at the end of the 8-m-long crane boom. If \(\theta = 30^{\circ}\), determine the placement x of the hook at B so that this force creates a maximum moment about point O. What is this moment?

The towline exerts a force of P = 6 kN at the end of the 8-m-long crane boom. If x = 10 m, determine the position \(\theta\) of the boom so that this force creates a maximum moment about point O. What is this moment?

The 20-N horizontal force acts on the handle of the socket wrench. What is the moment of this force about point B. Specify the coordinate direction angles \(\alpha\), \(\beta\), \(\gamma\) of the moment axis.

The 20-N horizontal force acts on the handle of the socket wrench. Determine the moment of this force about point O. Specify the coordinate direction angles \(\alpha\), \(\beta\), \(\gamma\) of the moment axis.

Two men exert forces of F = 80 lb and P = 50 lb on the ropes. Determine the moment of each force about A. Which way will the pole rotate, clockwise or counterclockwise?

If the man at B exerts a force of P = 30 lb on his rope, determine the magnitude of the force F the man at C must exert to prevent the pole from rotating, i.e., so the resultant moment about A of both forces is zero.

The torque wrench ABC is used to measure the moment or torque applied to a bolt when the bolt is located at A and a force is applied to the handle at C. The mechanic reads the torque on the scale at B. If an extension AO of length d is used on the wrench, determine the required scale reading if the desired torque on the bolt at O is to be M.

The tongs are used to grip the ends of the drilling pipe P. Determine the torque (moment) \(M_P\) that the applied force F = 150 lb exerts on the pipe about point P as a function of \(\theta\). Plot this moment \(M_P\) versus \(\theta\) for \(0\ \leq\ \theta\ \leq\ 90^{\circ}\).

The tongs are used to grip the ends of the drilling pipe P. If a torque (moment) of \(M_P = 800\ lb \cdot ft\) is needed at P to turn the pipe, determine the cable force F that must be applied to the tongs. Set \(\theta = 30^{\circ}\).

The handle of the hammer is subjected to the force of F = 20 lb. Determine the moment of this force about the point A.

In order to pull out the nail at B, the force F exerted on the handle of the hammer must produce a clockwise moment of \(500\ lb \cdot in\). about point A. Determine the required magnitude of force F.

Old clocks were constructed using a fusee B to drive the gears and watch hands. The purpose of the fusee is to increase the leverage developed by the mainspring A as it uncoils and thereby loses some of its tension. The mainspring can develop a torque (moment) \(T_s = k \theta\), where \(k = 0.015\ N \cdot m/rad\) is the torsional stiffness and \(\theta\) is the angle of twist of the spring in radians. If the torque \(T_f\) developed by the fusee is to remain constant as the mainspring winds down, and x = 10 mm when \(\theta = 4\ rad\), determine the required radius of the fusee when \(\theta = 3\ rad\).

The tower crane is used to hoist the 2-Mg load upward at constant velocity. The 1.5-Mg jib BD, 0.5-Mg jib BC, and 6-Mg counterweight C have centers of mass at \(G_1\), \(G_2\), and \(G_3\), respectively. Determine the resultant moment produced by the load and the weights of the tower crane jibs about point A and about point B.

The tower crane is used to hoist a 2-Mg load upward at constant velocity. The 1.5-Mg jib BD and 0.5-Mg jib BC have centers of mass at \(G_1\) and \(G_2\), respectively. Determine the required mass of the counterweight C so that the resultant moment produced by the load and the weight of the tower crane jibs about point A is zero. The center of mass for the counterweight is located at \(G_3\).

If the 1500-lb boom AB, the 200-lb cage BCD, and the 175-lb man have centers of gravity located at points \(G_1\), \(G_2\), and \(G_3\), respectively, determine the resultant moment produced by each weight about point A.

If the 1500-lb boom AB, the 200-lb cage BCD, and the 175-lb man have centers of gravity located at points \(G_1\), \(G_2\), and \(G_3\), respectively, determine the resultant moment produced by all the weights about point A.

Determine the moment of the force F about point O. Express the result as a Cartesian vector.

Determine the moment of the force F about point P. Express the result as a Cartesian vector.

The force \(F = \{400i - 100j - 700k\}\ lb\) acts at the end of the beam. Determine the moment of this force about point O.

The force \(F = \{400i - 100j - 700k\}\ lb\) acts at the end of the beam. Determine the moment of this force about point A.

Determine the moment of the force F about point P. Express the result as a Cartesian vector.

The pipe assembly is subjected to the force of \(F = \{600i + 800j - 500k\}\ N\). Determine the moment of this force about point A.

The pipe assembly is subjected to the force of \(F = \{600i + 800j - 500k\}\ N\). Determine the moment of this force about point B.

Determine the moment of the force of F = 600 N about point A.

Determine the smallest force F that must be applied along the rope in order to cause the curved rod, which has a radius of 4 m, to fail at the support A. This requires a moment of \(M = 1500\ N \cdot m\) to be developed at A.

Determine the coordinate direction angles \(\alpha\), \(\beta\), \(\gamma\) of force F, so that the moment of F about O is zero.

Determine the moment of force F about point O. The force has a magnitude of 800 N and coordinate direction angles of \(\alpha = 60^{\circ}\), \(\beta = 120^{\circ}\), \(\gamma = 45^{\circ}\). Express the result as a Cartesian vector.

Determine the moment of the force F about the door hinge at A. Express the result as a Cartesian vector.

Determine the moment of the force F about the door hinge at B. Express the result as a Cartesian vector.

The curved rod has a radius of 5 ft. If a force of 60 lb acts at its end as shown, determine the moment of this force about point C.

Determine the smallest force F that must be applied along the rope in order to cause the curved rod, which has a radius of 5 ft, to fail at the support C. This requires a moment of \(M = 80\ lb \cdot ft\) to be developed at C.

A 20-N horizontal force is applied perpendicular to the handle of the socket wrench. Determine the magnitude and the coordinate direction angles of the moment created by this force about point O.

The pipe assembly is subjected to the 80-N force. Determine the moment of this force about point A.

The pipe assembly is subjected to the 80-N force. Determine the moment of this force about point B.

A force of \(F = \{6i - 2j + 1k\}\ kN\) produces a moment of \(M_O = \{4i + 5j - 14k\}\ kN \cdot m\) about the origin, point O. If the force acts at a point having an x coordinate of x = 1 m, determine the y and z coordinates. Note: The figure shows F and \(M_O\) in an arbitrary position.

The force \(F = \{6i + 8j + 10k\}\ N\) creates a moment about point O of \(M_O = \{-14i + 8j + 2k\}\ N \cdot m\). If the force passes through a point having an x coordinate of 1 m, determine the y and z coordinates of the point. Also, realizing that \(M_O = Fd\), determine the perpendicular distance d from point O to the line of action of F. Note: The figure shows F and \(M_O\) in an arbitrary position.

A force F having a magnitude of F = 100 N acts along the diagonal of the parallelepiped. Determine the moment of F about the point A, using \(M_A = r_B \times F\) and \(M_A = r_C \times F\).

Force F acts perpendicular to the inclined plane. Determine the moment produced by F about point A. Express the result as a Cartesian vector.

Force F acts perpendicular to the inclined plane. Determine the moment produced by F about point B. Express the result as a Cartesian vector.

Strut AB of the 1-m-diameter hatch door exerts a force of 450 N on point B. Determine the moment of this force about point O.

Using a ring collar, the 75-N force can act in the vertical plane at various angles \(\theta\). Determine the magnitude of the moment it produces about point A, plot the result of M (ordinate) versus \(\theta\) (abscissa) for \(0^{\circ}\ \leq\ \theta\ \leq\ 180^{\circ}\), and specify the angles that give the maximum and minimum moment.

The lug nut on the wheel of the automobile is to be removed using the wrench and applying the vertical force of F = 30 N at A. Determine if this force is adequate, provided \(14\ N \cdot m\) of torque about the x axis is initially required to turn the nut. If the 30-N force can be applied at A in any other direction, will it be possible to turn the nut?

Solve Prob. 4–52 if the cheater pipe AB is slipped over the handle of the wrench and the 30-N force can be applied at any point and in any direction on the assembly.

The A-frame is being hoisted into an upright position by the vertical force of F = 80 lb. Determine the moment of this force about the y’ axis passing through points A and B when the frame is in the position shown.

The A-frame is being hoisted into an upright position by the vertical force of F = 80 lb. Determine the moment of this force about the x axis when the frame is in the position shown.

Determine the magnitude of the moments of the force F about the x, y, and z axes. Solve the problem (a) using a Cartesian vector approach and (b) using a scalar approach.

Determine the moment of this force F about an axis extending between A and C. Express the result as a Cartesian vector.

The board is used to hold the end of a four-way lug wrench in the position shown when the man applies a force of F = 100 N. Determine the magnitude of the moment produced by this force about the x axis. Force F lies in a vertical plane.

The board is used to hold the end of a four-way lug wrench in position. If a torque of \(30\ N \cdot m\) about the x axis is required to tighten the nut, determine the required magnitude of the force F that the man’s foot must apply on the end of the wrench in order to turn it. Force F lies in a vertical plane.

The A-frame is being hoisted into an upright position by the vertical force of F = 80 lb. Determine the moment of this force about the y axis when the frame is in the position shown.

Determine the magnitude of the moment of the force \(F = \{50i - 20j - 80k\}\ N\) about the base line AB of the tripod.

Determine the magnitude of the moment of the force \(F = \{50i - 20j - 80k\}\ N\) about the base line BC of the tripod.

Determine the magnitude of the moment of the force \(F = \{50i - 20j - 80k\}\ N\) about the base line CA of the tripod.

A horizontal force of \(F = \{-50i\}\ N\) is applied perpendicular to the handle of the pipe wrench. Determine the moment that this force exerts along the axis OA (z axis) of the pipe assembly. Both the wrench and pipe assembly, OABC, lie in the y-z plane. Suggestion: Use a scalar analysis.

Determine the magnitude of the horizontal force F = -Fi acting on the handle of the wrench so that this force produces a component of the moment along the OA axis (z axis) of the pipe assembly of \(M_z = \{4k\}\ N \cdot m\). Both the wrench and the pipe assembly, OABC, lie in the y-z plane. Suggestion: Use a scalar analysis.

The force of F = 30 N acts on the bracket as shown. Determine the moment of the force about the a-a axis of the pipe if \(\alpha = 60^{\circ}\), \(\beta = 60^{\circ}\), and \(\gamma = 45^{\circ}\). Also, determine the coordinate direction angles of F in order to produce the maximum moment about the a-a axis. What is this moment?

A clockwise couple \(M = 5\ N \cdot m\) is resisted by the shaft of the electric motor. Determine the magnitude of the reactive forces -R and R which act at supports A and B so that the resultant of the two couples is zero.

A twist of \(4\ N \cdot m\) is applied to the handle of the screwdriver. Resolve this couple moment into a pair of couple forces F exerted on the handle and P exerted on the blade.

If the resultant couple of the three couples acting on the triangular block is to be zero, determine the magnitude of forces F and P.

Two couples act on the beam. If F = 125 lb, determine the resultant couple moment.

Two couples act on the beam. Determine the magnitude of F so that the resultant couple moment is \(450\ lb \cdot ft\), counterclockwise. Where on the beam does the resultant couple moment act?

Determine the magnitude of the couple forces F so that the resultant couple moment on the crank is zero.

The ends of the triangular plate are subjected to three couples. Determine the magnitude of the force F so that the resultant couple moment is \(400\ N \cdot m\) clockwise.

The man tries to open the valve by applying the couple forces of F = 75 N to the wheel. Determine the couple moment produced.

If the valve can be opened with a couple moment of \(25\ N \cdot m\), determine the required magnitude of each couple force which must be applied to the wheel.

Determine the magnitude of F so that the resultant couple moment is \(12\ kN \cdot m\), counterclockwise. Where on the beam does the resultant couple moment act?

Two couples act on the beam as shown. If F = 150 lb, determine the resultant couple moment.

Two couples act on the beam as shown. Determine the magnitude of F so that the resultant couple moment is \(300\ lb \cdot ft\) counterclockwise. Where on the beam does the resultant couple act?

Two couples act on the frame. If the resultant couple moment is to be zero, determine the distance d between the 80-lb couple forces.

Two couples act on the frame. If d = 4 ft, determine the resultant couple moment. Compute the result by resolving each force into x and y components and (a) finding the moment of each couple (Eq. 4–13) and (b) summing the moments of all the force components about point A.

Two couples act on the frame. If d = 4 ft, determine the resultant couple moment. Compute the result by resolving each force into x and y components and (a) finding the moment of each couple (Eq. 4–13) and (b) summing the moments of all the force components about point B.

Express the moment of the couple acting on the pipe assembly in Cartesian vector form. What is the magnitude of the couple moment?

If \(M_1 = 180\ lb \cdot ft\), \(M_2 = 90\ lb \cdot ft\), and \(M_3 = 120\ lb \cdot ft\), determine the magnitude and coordinate direction angles of the resultant couple moment.

Determine the magnitudes of couple moments \(M_1\), \(M_2\), and \(M_3\) so that the resultant couple moment is zero.

The gears are subjected to the couple moments shown. Determine the magnitude and coordinate direction angles of the resultant couple moment.

Determine the required magnitude of the couple moments \(M_2\) and \(M_3\) so that the resultant couple moment is zero.

Determine the resultant couple moment of the two couples that act on the assembly. Specify its magnitude and coordinate direction angles.

Express the moment of the couple acting on the frame in Cartesian vector form. The forces are applied perpendicular to the frame. What is the magnitude of the couple moment? Take F = 50 N.

In order to turn over the frame, a couple moment is applied as shown. If the component of this couple moment along the x axis is \(M_x = \{-20i\}\ N \cdot m\), determine the magnitude F of the couple forces.

Express the moment of the couple acting on the pipe in Cartesian vector form. What is the magnitude of the couple moment? Take F = 125 N.

If the couple moment acting on the pipe has a magnitude of \(300\ N \cdot m\), determine the magnitude F of the forces applied to the wrenches.

If F = 80 N, determine the magnitude and coordinate direction angles of the couple moment. The pipe assembly lies in the x–y plane.

If the magnitude of the couple moment acting on the pipe assembly is \(50\ N \cdot m\), determine the magnitude of the couple forces applied to each wrench. The pipe assembly lies in the x–y plane.

Express the moment of the couple acting on the rod in Cartesian vector form. What is the magnitude of the couple moment?

If \(F_1 = 100\ N\), \(F_2 = 120\ N\), and \(F_3 = 80\ N\), determine the magnitude and coordinate direction angles of the resultant couple moment.

Determine the required magnitude of \(F_1\), \(F_2\), and \(F_3\) so that the resultant couple moment is \((M_c)_R = [50i - 45j - 20k]\ N \cdot m\).

Replace the force system by an equivalent resultant force and couple moment at point O.

Replace the force system by an equivalent resultant force and couple moment at point P.

Replace the force system acting on the beam by an equivalent force and couple moment at point A.

Replace the force system acting on the beam by an equivalent force and couple moment at point B.

Replace the loading system acting on the beam by an equivalent resultant force and couple moment at point O.

Replace the loading system acting on the post by an equivalent resultant force and couple moment at point A.

Replace the loading system acting on the post by an equivalent resultant force and couple moment at point B.

Replace the force system acting on the post by a resultant force and couple moment at point O.

Replace the force system acting on the frame by an equivalent resultant force and couple moment acting at point A.

The forces \(F_1 = \{-4i + 2j - 3k\}\ kN\) and \(F_2 = \{3i - 4j - 2k\}\ kN\) act on the end of the beam. Replace these forces by an equivalent force and couple moment acting at point O.

A biomechanical model of the lumbar region of the human trunk is shown. The forces acting in the four muscle groups consist of \(F_R = 35\ N\) for the rectus, \(F_O = 45\ N\) for the oblique, \(F_L = 23\ N\) for the lumbar latissimus dorsi, and \(F_E = 32\ N\) for the erector spinae. These loadings are symmetric with respect to the y–z plane. Replace this system of parallel forces by an equivalent force and couple moment acting at the spine, point O. Express the results in Cartesian vector form.

Replace the force system by an equivalent resultant force and couple moment at point O. Take \(F_3 = \{-200i + 500j - 300k\}\ N\).

Replace the loading by an equivalent resultant force and couple moment at point O.

Replace the force of F = 80 N acting on the pipe assembly by an equivalent resultant force and couple moment at point A.

The belt passing over the pulley is subjected to forces \(F_1\) and \(F_2\), each having a magnitude of 40 N. \(F_1\) acts in the -k direction. Replace these forces by an equivalent force and couple moment at point A. Express the result in Cartesian vector form. Set \(\theta = 0^{\circ}\) so that \(F_2\) acts in the -j direction.

The belt passing over the pulley is subjected to two forces \(F_1\) and \(F_2\), each having a magnitude of 40 N. \(F_1\) acts in the -k direction. Replace these forces by an equivalent force and couple moment at point A. Express the result in Cartesian vector form. Take \(\theta = 45^{\circ}\).

The weights of the various components of the truck are shown. Replace this system of forces by an equivalent resultant force and specify its location measured from B.

The weights of the various components of the truck are shown. Replace this system of forces by an equivalent resultant force and specify its location measured from point A.

Replace the three forces acting on the shaft by a single resultant force. Specify where the force acts, measured from end A.

Replace the three forces acting on the shaft by a single resultant force. Specify where the force acts, measured from end B.

Replace the loading acting on the beam by a single resultant force. Specify where the force acts, measured from end A.

Replace the loading acting on the beam by a single resultant force. Specify where the force acts, measured from B.

Replace the loading on the frame by a single resultant force. Specify where its line of action intersects a vertical line along member AB, measured from A.

Replace the loading on the frame by a single resultant force. Specify where its line of action intersects a vertical line along member AB, measured from A.

Replace the loading on the frame by a single resultant force. Specify where its line of action intersects a horizontal line along member CB, measured from end C.

Replace the force system acting on the post by a resultant force, and specify where its line of action intersects the post AB measured from point A.

Replace the force system acting on the post by a resultant force, and specify where its line of action intersects the post AB measured from point B.

Replace the parallel force system acting on the plate by a resultant force and specify its location on the x-z plane.

Replace the force and couple system acting on the frame by an equivalent resultant force and specify where the resultant’s line of action intersects member AB, measured from A.

Replace the force and couple system acting on the frame by an equivalent resultant force and specify where the resultant’s line of action intersects member BC, measured from B.

If \(F_A = 7\ kN\) and \(F_B = 5\ kN\), represent the force system acting on the corbels by a resultant force, and specify its location on the x–y plane.

Determine the magnitudes of \(F_A\) and \(F_B\) so that the resultant force passes through point O of the column.

The tube supports the four parallel forces. Determine the magnitudes of forces \(F_C\) and \(F_D\) acting at C and D so that the equivalent resultant force of the force system acts through the midpoint O of the tube.

The building slab is subjected to four parallel column loadings. Determine the equivalent resultant force and specify its location (x, y) on the slab. Take \(F_1 = 8\ kN\) and \(F_2 = 9\ kN\).

The building slab is subjected to four parallel column loadings. Determine \(F_1\) and \(F_2\) if the resultant force acts through point (12 m, 10 m).

If \(F_A = 40\ kN\) and \(F_B = 35\ kN\), determine the magnitude of the resultant force and specify the location of its point of application (x, y) on the slab.

If the resultant force is required to act at the center of the slab, determine the magnitude of the column loadings \(F_A\) and \(F_B\) and the magnitude of the resultant force.

Replace the two wrenches and the force, acting on the pipe assembly, by an equivalent resultant force and couple moment at point O.

Replace the force system by a wrench and specify the magnitude of the force and couple moment of the wrench and the point where the wrench intersects the x–z plane.

Replace the five forces acting on the plate by a wrench. Specify the magnitude of the force and couple moment for the wrench and the point P(x, z) where the wrench intersects the x–z plane.

Replace the three forces acting on the plate by a wrench. Specify the magnitude of the force and couple moment for the wrench and the point P(x, y) where the wrench intersects the plate.

Replace the loading by an equivalent resultant force and couple moment acting at point O.

Replace the distributed loading with an equivalent resultant force, and specify its location on the beam measured from point O.

Replace the loading by an equivalent resultant force and specify its location on the beam, measured from point A.

Currently eighty-five percent of all neck injuries are caused by rear-end car collisions. To alleviate this problem, an automobile seat restraint has been developed that provides additional pressure contact with the cranium. During dynamic tests the distribution of load on the cranium has been plotted and shown to be parabolic. Determine the equivalent resultant force and its location, measured from point A.

Replace the distributed loading by an equivalent resultant force, and specify its location on the beam, measured from the pin at A.

Replace this loading by an equivalent resultant force and specify its location, measured from point O.

The distribution of soil loading on the bottom of a building slab is shown. Replace this loading by an equivalent resultant force and specify its location, measured from point O.

Replace the loading by an equivalent resultant force and couple moment acting at point O.

Replace the distributed loading by an equivalent resultant force and couple moment acting at point A.

Determine the length b of the triangular load and its position a on the beam such that the equivalent resultant force is zero and the resultant couple moment is \(8\ kN \cdot m\) clockwise.

The form is used to cast a concrete wall having a width of 5 m. Determine the equivalent resultant force the wet concrete exerts on the form AB if the pressure distribution due to the concrete can be approximated as shown. Specify the location of the resultant force, measured from point B.

If the soil exerts a trapezoidal distribution of load on the bottom of the footing, determine the intensities \(w_1\) and \(w_2\) of this distribution needed to support the column loadings.

Replace the loading by an equivalent force and couple moment acting at point O.

Replace the loading by a single resultant force, and specify the location of the force measured from point O.

Replace the loading by an equivalent resultant force and couple moment acting at point A.

Replace the loading by a single resultant force, and specify its location on the beam measured from point A.

Replace the distributed loading by an equivalent resultant force and specify where its line of action intersects a horizontal line along member AB, measured from A.

Replace the distributed loading by an equivalent resultant force and specify where its line of action intersects a vertical line along member BC, measured from C.

Determine the length b of the triangular load and its position a on the beam such that the equivalent resultant force is zero and the resultant couple moment is 8 kN # m clockwise.

Determine the equivalent resultant force and couple moment at point O.

Determine the magnitude of the equivalent resultant force and its location, measured from point O.

The distributed load acts on the shaft as shown. Determine the magnitude of the equivalent resultant force and specify its location, measured from the support, A.

Replace the distributed loading with an equivalent resultant force, and specify its location on the beam measured from point A.

Replace the loading by an equivalent resultant force and couple moment acting at point O.

Wet concrete exerts a pressure distribution along the wall of the form. Determine the resultant force of this distribution and specify the height h where the bracing strut should be placed so that it lies through the line of action of the resultant force. The wall has a width of 5 m.