The beam is to be hoisted using two chains. If the

Determine the magnitude of the resultant force \(F_R = F_1 + F_2\) and its direction, measured counterclockwise from the positive x axis.

If \(\theta = 60^{\circ}\) and \(F = 450\ N\), determine the magnitude of the resultant force and its direction, measured counterclockwise from the positive x axis.

If the magnitude of the resultant force is to be 500 N, directed along the positive y axis, determine the magnitude of force F and its direction \(\theta\).

Determine the magnitude of the resultant force \(F_R = F_1 + F_2\) and its direction, measured clockwise from the positive u axis.

Resolve the force \(F_1\) into components acting along the u and v axes and determine the magnitudes of the components.

Resolve the force \(F_2\) into components acting along the u and v axes and determine the magnitudes of the components.

The vertical force F acts downward at A on the twomembered frame. Determine the magnitudes of the two components of F directed along the axes of AB and AC. Set F = 500 N.

Solve Prob. 2–7 with F = 350 lb.

Resolve \(F_1\) into components along the u and v axes and determine the magnitudes of these components.

Resolve \(F_2\) into components along the u and v axes and determine the magnitudes of these components.

The force acting on the gear tooth is F = 20 lb. Resolve this force into two components acting along the lines aa and bb.

The component of force F acting along line aa is required to be 30 lb. Determine the magnitude of F and its component along line bb.

Force F acts on the frame such that its component acting along member AB is 650 lb, directed from B towards A, and the component acting along member BC is 500 lb, directed from B towards C. Determine the magnitude of F and its direction \(\theta\). Set \(\phi = 60^{\circ}\).

Force F acts on the frame such that its component acting along member AB is 650 lb, directed from B towards A. Determine the required angle \(\phi (0^{\circ}\ \leq\ \phi\ \leq\ 90^{\circ})\) and the component acting along member BC. Set F = 850 lb and \(\theta = 30^{\circ}\).

The plate is subjected to the two forces at A and B as shown. If \(\theta = 60^{\circ}\), determine the magnitude of the resultant of these two forces and its direction measured clockwise from the horizontal.

Determine the angle \(\theta\) for connecting member A to the plate so that the resultant force of \(F_A\) and \(F_B\) is directed horizontally to the right. Also, what is the magnitude of the resultant force?

Determine the design angle \(\theta (0^{\circ}\ \theta\ \theta\ \leq\ 90^{\circ})\) for strut AB so that the 400-lb horizontal force has a component of 500 lb directed from A towards C. What is the component of force acting along member AB? Take \(\phi = 40^{\circ}\).

Determine the design angle \(\phi (0^{\circ}\ \leq\ \phi\ \leq\ 90^{\circ})\) between struts AB and AC so that the 400-lb horizontal force has a component of 600 lb which acts up to the left, in the same direction as from B towards A. Take \(\theta = 30^{\circ}\).

Determine the magnitude and direction of the resultant \(F_R = F_1 + F_2 + F_3\) of the three forces by first finding the resultant \(F’ = F_1 + F_2\) and then forming \(F_R = F + F_3\).

Determine the magnitude and direction of the resultant \(F_R = F_1 + F_2 + F_3\) of the three forces by first finding the resultant \(F’ = F_2 + F_3\) and then forming \(F_R = F’ + F_1\).

Two forces act on the screw eye. If \(F_1 = 400\ N\) and \(F_2 = 600\ N\), determine the angle \(\theta (0^{\circ}\ \leq\ \theta\ \leq\ 180^{\circ})\) between them, so that the resultant force has a magnitude of \(F_R = 800\ N\).

Two forces \(F_1\) and \(F_2\) act on the screw eye. If their lines of action are at an angle \(\theta\) apart and the magnitude of each force is \(F_1 = F_2 = F\), determine the magnitude of the resultant force \(F_R\) and the angle between \(F_R\) and \(F_1\).

Two forces act on the screw eye. If F = 600 N, determine the magnitude of the resultant force and the angle \(\theta\) if the resultant force is directed vertically upward.

Two forces are applied at the end of a screw eye in order to remove the post. Determine the angle \(\theta (0^{\circ}\ \leq\ \theta\ \leq\ 90^{\circ})\) and the magnitude of force F so that the resultant force acting on the post is directed vertically upward and has a magnitude of 750 N.

The chisel exerts a force of 20 lb on the wood dowel rod which is turning in a lathe. Resolve this force into components acting (a) along the n and t axes and (b) along the x and y axes.

The beam is to be hoisted using two chains. Determine the magnitudes of forces \(F_A\) and \(F_B\) acting on each chain in order to develop a resultant force of 600 N directed along the positive y axis. Set \(\theta = 45^{\circ}\).

The beam is to be hoisted using two chains. If the resultant force is to be 600 N directed along the positive y axis, determine the magnitudes of forces \(F_A\) and \(F_B\) acting on each chain and the angle \(\theta\) of \(F_B\) so that the magnitude of \(F_B\) is a minimum. \(F_A\) acts at \(30^{\circ}\) from the y axis, as shown.

If the resultant force of the two tugboats is 3 kN, directed along the positive x axis, determine the required magnitude of force \(F_B\) and its direction \(\theta\).

If \(F_B = 3\ kN\) and \(\theta = 45^{\circ}\), determine the magnitude of the resultant force of the two tugboats and its direction measured clockwise from the positive x axis.

If the resultant force of the two tugboats is required to be directed towards the positive x axis, and \(F_B\) is to be a minimum, determine the magnitude of \(F_R\) and \(F_B\) and the angle \(\theta\).

Three chains act on the bracket such that they create a resultant force having a magnitude of 500 lb. If two of the chains are subjected to known forces, as shown, determine the angle \(\theta\) of the third chain measured clockwise from the positive x axis, so that the magnitude of force F in this chain is a minimum . All forces lie in the x – y plane. What is the magnitude of F? Hint : First find the resultant of the two known forces. Force F acts in this direction.

Determine the x and y components of the 800-lb force.

Determine the magnitude of the resultant force and its direction, measured counterclockwise from the positive x axis.

Resolve \(F_1\) and \(F_2\) into their x and y components.

Determine the magnitude of the resultant force and its direction measured counterclockwise from the positive x axis.

Resolve each force acting on the gusset plate into its x and y components, and express each force as a Cartesian vector.

Determine the magnitude of the resultant force acting on the plate and its direction, measured counterclockwise from the positive x axis.

Express each of the three forces acting on the column in Cartesian vector form and compute the magnitude of the resultant force.

Resolve each force acting on the support into its x and y components, and express each force as a Cartesian vector.

Determine the magnitude of the resultant force and its direction \(\theta\), measured counterclockwise from the positive x axis.

Determine the magnitude of the resultant force and its direction measured counterclockwise from the positive x axis.

Determine the magnitude and orientation \(\theta\) of \(F_B\) so that the resultant force is directed along the positive y axis and has a magnitude of 1500 N.

Determine the magnitude and orientation, measured counterclockwise from the positive y axis, of the resultant force acting on the bracket, if \(F_B = 600\ N\) and \(\theta = 20^{\circ}\).

The magnitude of the resultant force acting on the bracket is to be 400 N. Determine the magnitude of \(F_1\) if \(\phi = 30^{\circ}\).

If the resultant force acting on the bracket is to be directed along the positive u axis, and the magnitude of \(F_1\) is required to be minimum, determine the magnitudes of the resultant force and \(F_1\).

If the magnitude of the resultant force acting on the bracket is 600 N, directed along the positive u axis, determine the magnitude of F and its direction \(\phi\).

Determine the magnitude and direction \(\theta\) of the resultant force \(F_R\). Express the result in terms of the magnitudes of the components \(F_1\) and \(F_2\) and the angle \(\phi\).

If \(F_1 = 600\ N\) and \(\phi = 30^{\circ}\), determine the magnitude of the resultant force acting on the eyebolt and its direction, measured clockwise from the positive x axis.

If the magnitude of the resultant force acting on the eyebolt is 600 N and its direction measured clockwise from the positive x axis is \(\theta = 30^{\circ}\), determine the magnitude of \(F_1\) and the angle \(\phi\).

Determine the magnitude of \(F_1\) and its direction \(\theta\) so that the resultant force is directed vertically upward and has a magnitude of 800 N.

Determine the magnitude and direction measured counterclockwise from the positive x axis of the resultant force of the three forces acting on the ring A. Take \(F_1 = 500\ N\) and \(\theta = 20^{\circ}\).

Determine the magnitude of force F so that the resultant \(F_R\) of the three forces is as small as possible. What is the minimum magnitude of \(F_R\)?

Determine the magnitude of force F so that the resultant force of the three forces is as small as possible. What is the magnitude of the resultant force?

Three forces act on the bracket. Determine the magnitude and direction \(\theta\) of \(F_1\) so that the resultant force is directed along the positive x’ axis and has a magnitude of 1 kN.

If \(F_1 = 300\ N\) and \(\theta = 20^{\circ}\), determine the magnitude and direction, measured counterclockwise from the x’ axis, of the resultant force of the three forces acting on the bracket.

Three forces act on the bracket. Determine the magnitude and direction \(\theta\) of \(F_2\) so that the resultant force is directed along the positive u axis and has a magnitude of 50 lb.

If \(F_2 = 150\ lb\) and \(\theta = 55^{\circ}\), determine the magnitude and direction measured clockwise from the positive x axis, of the resultant force of the three forces acting on the bracket.

If the magnitude of the resultant force acting on the bracket is to be 450 N directed along the positive u axis, determine the magnitude of \(F_1\) and its direction \(\phi\).

If the resultant force acting on the bracket is required to be a minimum, determine the magnitude of \(F_1\) and the resultant force. Set \(\phi = 30^{\circ}\).

The stock mounted on the lathe is subjected to a force of 60 N. Determine the coordinate direction angle \(\beta\) and express the force as a Cartesian vector.

Determine the magnitude and coordinate direction angles of the resultant force and sketch this vector on the coordinate system.

Specify the coordinate direction angles of \(F_1\) and \(F_2\) and express each force as a Cartesian vector.

The bolt is subjected to the force F, which has components acting along the x, y, z axes as shown. If the magnitude of F is 80 N, and \(\alpha = 60^{\circ}\) and \(\gamma = 45^{\circ}\), determine the magnitudes of its components.

Determine the magnitude and coordinate direction angles of \(\mathbf{F}_{1}=\{60 \mathbf{i}-50 \mathbf{j}+40 \mathbf{k}\}\ \mathrm{N}\) and \(\mathbf{F}_{2}=\{-40 \mathbf{i}-85 \mathbf{j}+30 \mathbf{k}\}\ \mathrm{N}\). Sketch each force on an x, y, z reference frame.

The cable at the end of the crane boom exerts a force of 250 lb on the boom as shown. Express F as a Cartesian vector.

Express each force acting on the pipe assembly in Cartesian vector form.

Determine the magnitude and the direction of the resultant force acting on the pipe assembly.

Express each force as a Cartesian vector.

Determine the magnitude and coordinate direction angles of the resultant force acting on the hook.

The beam is subjected to the two forces shown. Express each force in Cartesian vector form and determine the magnitude and coordinate direction angles of the resultant force.

If the resultant force acting on the bracket is directed along the positive y axis, determine the magnitude of the resultant force and the coordinate direction angles of F so that \(\beta\ <\ 90^{\circ}\).

A force F is applied at the top of the tower at A. If it acts in the direction shown such that one of its components lying in the shaded y-z plane has a magnitude of 80 lb, determine its magnitude F and coordinate direction angles \(\alpha\), \(\beta\), \(\gamma\).

The spur gear is subjected to the two forces caused by contact with other gears. Express each force as a Cartesian vector.

The spur gear is subjected to the two forces caused by contact with other gears. Determine the resultant of the two forces and express the result as a Cartesian vector.

Determine the coordinate direction angles of force \(F_1\).

Determine the magnitude and coordinate direction angles of the resultant force acting on the eyebolt.

The cables attached to the screw eye are subjected to the three forces shown. Express each force in Cartesian vector form and determine the magnitude and coordinate direction angles of the resultant force.

Three forces act on the ring. If the resultant force \(F_R\) has a magnitude and direction as shown, determine the magnitude and the coordinate direction angles of force \(F_3\).

Determine the coordinate direction angles of \(F_1\) and \(F_R\).

If the coordinate direction angles for \(F_3\) are \(\alpha_3 = 120^{\circ}\), \(\beta_3 = 45^{\circ}\) and \(\gamma_3 = 60^{\circ}\), determine the magnitude and coordinate direction angles of the resultant force acting on the eyebolt.

If the coordinate direction angles for \(F_3\) are \(\alpha_3 = 120^{\circ}\), \(\beta_3 = 45^{\circ}\), and \(\gamma_3 = 60^{\circ}\), determine the magnitude and coordinate direction angles of the resultant force acting on the eyebolt.

If the direction of the resultant force acting on the eyebolt is defined by the unit vector \(\mathbf{u}_{F_{R}}=\cos\ 30^{\circ} \mathbf{j}+\sin\ 30^{\circ} \mathbf{k}\), determine the coordinate direction angles of \(F_3\) and the magnitude of \(F_R\).

The bracket is subjected to the two forces shown. Express each force in Cartesian vector form and then determine the resultant force \(F_R\). Find the magnitude and coordinate direction angles of the resultant force.

The pole is subjected to the force F, which has components acting along the x, y, z axes as shown. If the magnitude of F is 3 kN, \(\beta = 30^{\circ}\), and \(\gamma = 75^{\circ}\), determine the magnitudes of its three components.

The pole is subjected to the force F which has components \(F_x = 1.5\ kN\) and \(F_z = 1.25\ kN\). If \(\beta = 75^{\circ}\), determine the magnitudes of F and \(\mathbf{F}_y\).

Express the position vector r in Cartesian vector form; then determine its magnitude and coordinate direction angles.

Determine the lengths of wires AD, BD, and CD. The ring at D is midway between A and B.

Determine the length of member AB of the truss by first establishing a Cartesian position vector from A to B and then determining its magnitude.

If \(\mathbf{F}=\{350 \mathbf{i}-250 \mathbf{j}-450 \mathbf{k}\}\ N\) and cable AB is 9 m long, determine the x, y, z coordinates of point A.

Express \(\mathbf{F}_{B}\) and \(\mathbf{F}_{C}\) in Cartesian vector form.

Determine the magnitude and coordinate direction angles of the resultant force acting at A.

If \(\mathbf{F}_{B}=560\ N\) and \(\mathbf{F}_{C}=700\ N\), determine the magnitude and coordinate direction angles of the resultant force acting on the flag pole.

If \(\mathbf{F}_B = 700\ N\), and \(\mathbf{F}_C = 560\ N\), determine the magnitude and coordinate direction angles of the resultant force acting on the flag pole.

The tower is held in place by three cables. If the force of each cable acting on the tower is shown, determine the magnitude and coordinate direction angles \(\alpha\), \(\beta\), \(\gamma\) of the resultant force. Take x = 20 m, y = 15 m.

At a given instant, the position of a plane at A and a train at B are measured relative to a radar antenna at O. Determine the distance d between A and B at this instant. To solve the problem, formulate a position vector, directed from A to B, and then determine its magnitude.

The man pulls on the rope at C with a force of 70 lb which causes the forces \(\mathbf{F}_A\) and \(\mathbf{F}_C\) at B to have this same magnitude. Express each of these two forces as Cartesian vectors.

The man pulls on the rope at C with a force of 70 lb which causes the forces \(\mathbf{F}_A\) and \(\mathbf{F}_C\) at B to have this same magnitude. Determine the magnitude and coordinate direction angles of the resultant force acting at B.

The load at A creates a force of 60 lb in wire AB. Express this force as a Cartesian vector acting on A and directed toward B as shown.

Determine the magnitude and coordinate direction angles of the resultant force acting at point A.

The guy wires are used to support the telephone pole. Represent the force in each wire in Cartesian vector form. Neglect the diameter of the pole.

The two mooring cables exert forces on the stern of a ship as shown. Represent each force as as Cartesian vector and determine the magnitude and coordinate direction angles of the resultant.

Each of the four forces acting at E has a magnitude of 28 kN. Express each force as a Cartesian vector and determine the resultant force.

If the force in each cable tied to the bin is 70 lb, determine the magnitude and coordinate direction angles of the resultant force.

If the resultant of the four forces is \(\mathbf{F}_R=\{-360 \mathbf{k}\}\ lb\), determine the tension developed in each cable. Due to symmetry, the tension in the four cables is the same.

The pipe is supported at its ends by a cord AB. If the cord exerts a force of F = 12 lb on the pipe at A, express this force as a Cartesian vector.

The chandelier is supported by three chains which are concurrent at point O. If the force in each chain has a magnitude of 60 lb, express each force as a Cartesian vector and determine the magnitude and coordinate direction angles of the resultant force.

The chandelier is supported by three chains which are concurrent at point O. If the resultant force at O has a magnitude of 130 lb and is directed along the negative z axis, determine the force in each chain.

Determine the magnitude and coordinate direction angles of the resultant force. Set \(\mathbf{F}_B=630\ N\), \(\mathbf{F}_C=520\ N\) and \(\mathbf{F}_D=750\ N\), and x = 3 m and z = 3.5 m.

If the magnitude of the resultant force is 1300 N and acts along the axis of the strut, directed from point A towards O, determine the magnitudes of the three forces acting on the strut. Set x = 0 and z = 5.5 m.

The cable attached to the shear-leg derrick exerts a force on the derrick of F = 350 lb. Express this force as a Cartesian vector.

The window is held open by chain AB. Determine the length of the chain, and express the 50-lb force acting at A along the chain as a Cartesian vector and determine its coordinate direction angles.

Given the three vectors A, B, and D, show that \(\mathbf{A} \cdot (\mathbf{B} + \mathbf{D}) = (\mathbf{A} \cdot \mathbf{B}) +(\mathbf{A} \cdot \mathbf{D})\).

Determine the angle \(\theta\) between the edges of the sheet-metal bracket.

Determine the angle \(\theta\) between the sides of the triangular plate.

Determine the length of side BC of the triangular plate. Solve the problem by finding the magnitude of \(\mathbf{r}_{BC}\); then check the result by first finding \(\theta\), \(r_{AB}\), and \(r_{AC}\) and then use the cosine law.

Determine the magnitude of the projected component of force \(\mathbf{F}_{AB}\) acting along the z axis.

Determine the magnitude of the projected component of force \(\mathbf{F}_{AC}\) acting along the z axis.

Determine the projection of the force F along the pole.

Determine the angle \(\theta\) between the y axis of the pole and the wire AB.

Determine the magnitudes of the components of F = 600 N acting along and perpendicular to segment DE of the pipe assembly.

Determine the magnitude of the projection of force F = 600 N along the u axis.

Determine the angle \(\theta\) between cables AB and AC.

Determine the angle \(\phi\) between cable AC and strut AO.

Determine the projected component of force \(\mathbf{F}_{AB}\) along the axis of strut AO. Express the result as a Cartesian vector.

Determine the projected component of force \(\mathbf{F}_{AC}\) along the axis of strut AO. Express the result as a Cartesian vector.

Two cables exert forces on the pipe. Determine the magnitude of the projected component of \(\mathbf{F}_1\) along the line of action of \(\mathbf{F}_2\).

Determine the angle \(\theta\) between the two cables attached to the pipe.

Determine the magnitudes of the components of F acting along and perpendicular to segment BC of the pipe assembly.

Determine the magnitude of the projected component of F along AC. Express this component as a Cartesian vector.

Determine the angle \(\theta\) between the pipe segments BA and BC.

Determine the angles \(\theta\) and \(\phi\) made between the axes OA of the flag pole and AB and AC, respectively, of each cable.

The cables each exert a force of 400 N on the post. Determine the magnitude of the projected component of \(\mathbf{F}_1\) along the line of action of \(\mathbf{F}_2\).

Determine the angle \(\theta\) between the two cables attached to the post.

Determine the magnitudes of the components of force F = 90 lb acting parallel and perpendicular to diagonal AB of the crate.

The force \(\mathbf{F}=\{25 \mathbf{i}-50 \mathbf{j}+10 \mathbf{k}\}\ \mathrm{lb}\) acts at the end A of the pipe assembly. Determine the magnitude of the components \(\mathbf{F}_1\) and \(\mathbf{F}_2\) which act along the axis of AB and perpendicular to it.

Determine the components of F that act along rod AC and perpendicular to it. Point B is located at the midpoint of the rod.

Determine the components of F that act along rod AC and perpendicular to it. Point B is located 3 m along the rod from end C.

Determine the magnitudes of the projected components of the force F = 300 N acting along the x and y axes.

Determine the magnitude of the projected component of the force F = 300 N acting along line OA.