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Determine the position (x, y, 0) for fixing cable BA so that the resultant of the forces exerted on the pole is directed along its axis, from B toward O, and has magnitude FR. Also, what is the magnitude of force F3?

Given:
F1 = 500 N
F2 = 400 N
FR = 1000 N
a = 1 m
b = 2 m
c = 2 m
d = 3 m

The cord exerts a force F on the hook. If the cord is length L, determine the location x, y of the point of attachment B, and the height z of the hook.
Given:
F = (12 9 -8) lb
L = 8 ft
a = 2 ft

The cord exerts a force of magnitude F on the hook. If the cord length L, the distance z, and the x component of the force, Fx, are given, determine the location x, y of the point of attachment B of the cord to the ground.
Given:
F = 30 lb
L = 8 ft
z = 4 ft
Fx = 25 lb
a = 2 ft

Each of the four forces acting at E has magnitude F. Express each force as a Cartesian vector and determine the resultant force. Units used:
kN = 103 N
Given:
F = 28kN
a = 4 m
b = 6 m
c = 12 m

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 α, β, γ of the resultant force.

Units Used:
kN = 103 N
Given:
x = 20 m
a = 16 m
y = 15 m
b = 18 m
F1 = 600 N
c = 6 m
F2 = 400 N
d = 4 m
F3 = 800 N
e = 24 m

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

Given:
F = 60 lb
a = 6 ft
b = 4 ft
θ1 = 120 deg
θ2 = 120 deg

The chandelier is supported by three chains which are concurrent at point O. If the resultant force at O has magnitude FR and is directed along the negative z axis, determine the force in each chain assuming FA = FB = FC = F.
Given:
a = 6 ft
b = 4 ft
FR = 130 lb

Given the three vectors A, B, and D, show that A ∙ (B+D) = (A ∙ B) + (A ∙ D).
Cable BC exerts force F on the top of the flagpole. Determine the projection of this force along the z axis of the pole.
Given:
F = 28 N
a = 12 m
b = 6 m
c = 4 m

Determine the angle θ between the tails of the two vectors.

Given:
r1 = 9 m
r2 = 6 m
α = 60 deg
β = 45 deg
γ = 120 deg
φ = 30 deg
ε = 40 deg

Determine the magnitude of the projected component of r1 along r2, and the projection of r2 along r1.

Given:
r1 = 9 m
r2 = 6 m
α = 60 deg
β = 45 deg
γ = 120 deg
φ = 30 deg
ε = 40 deg

Determine the angles θ and φ between the wire segments.

Given:
a = 0.6
b = 0.8
c = 0.5
d = 0.2

Determine the angle θ between the two cords.

Given:
a = 3 m
b = 2 m
c = 6 m
d = 3 m
e = 4 m

Determine the angle θ between the two cables.

Given:
a = 8 ft
b = 10 ft
c = 8 ft
d = 10 ft
e = 4 ft
f = 6 ft
FAB = 12 lb

Determine the projected component of the force F acting in the direction of cable AC. Express the result as a Cartesian vector.
Given:
F = 12 lb
a = 8 ft
b = 10 ft
c = 8 ft
d = 10 ft
e = 4 ft
f = 6 ft

Determine the components of F that act along rod AC and perpendicular to it. Point B is located at the midpoint of the rod.
Given:
F = 600 N c = 4 m
a = 4 m d = 3 m
b = 6 m e = 4 m

Determine the components of F that act along rod AC and perpendicular to it. Point B is located a distance f along the rod from end C.
Given:
F = 600 N c = 4 m
a = 4 m d = 3 m
b = 6 m e = 4 m
f = 3 m

Determine the magnitude of the projected component of the length of cord OA along the Oa axis.

Given:
a = 10 ft
b = 5 ft
c = 15 ft
d = 5 ft
θ1 = 45 deg
θ2 = 60 deg

Force F acts at the end of the pipe. Determine the magnitudes of the components F1 and F2 which are directed along the pipe's axis and perpendicular to it.
Given:

Determine the projected component of the force F acting along the axis AB of the pipe.
Given:
F = 80 N
a = 4 m
b = 3 m
c = 12 m
d = 2 m
e = 6 m

Determine the angles θ and φ between the axis OA of the pole and each cable, AB and AC.

Given:
F1 = 50 N
F2 = 35 N
a = 1 m
b = 3 m
c = 2 m
d = 5 m
e = 4 m
f = 6 m
g = 4 m

The two cables exert the forces shown on the pole. Determine the magnitude of the projected component of each force acting along the axis OA of the pole.
Given:
F1 = 50 N
F2 = 35 N
a = 1 m
b = 3 m
c = 2 m
d = 5 m
e = 4 m
f = 6 m
g = 4 m

Force F is applied to the handle of the wrench. Determine the angle θ between the tail of the force and the handle AB.

Given:
a = 300 mm
b = 500 mm
F = 80 N
θ1 = 30 deg
θ2 = 45 deg

Two cables exert forces on the pipe. Determine the magnitude of the projected component of F1 along the line of action of F2.

Given:
F1 = 30 lb
β = 30 deg
F2 = 25 lb
γ = 60 deg
α = 30 deg
ε = 60 deg

Determine the angle θ between the two cables attached to the pipe.

Given:
F1 = 30 lb
β = 30 deg
F2 = 25 lb
γ = 60 deg
α = 30 deg
ε = 60 deg

Determine the angle θ between the two cables.

Given:
a = 7.5 ft
b = 2 ft
c = 3 ft
d = 2 ft
e = 3 ft
f = 3 ft
F1 = 60 lb
F2 = 30 lb

Determine the projection of the force F1 along cable AB. Determine the projection of the force F2 along cable AC.
Given:
a = 7.5 ft
b = 2 ft
c = 3 ft
d = 2 ft
e = 3 ft
f = 3 ft
F1 = 60 lb
F2 = 30 lb

Determine the angle θ between the edges of the sheet-metal bracket.

Given:
a = 50 mm
b = 300 mm
c = 250 mm
d = 400 mm

Determine the magnitude of the projected component of the force F acting along the axis BC of the pipe.
Given:
F = 100 lb
a = 2 ft
b = 8 ft
c = 6 ft
d = 4 ft
e = 2 ft

Determine the angle θ between pipe segments BA and BC.

Given:
F = 100 lb
a = 3 ft
b = 8 ft
c = 6 ft
d = 4 ft
e = 2 ft

Determine the angles θ and φ made between the axes OA of the flag pole and AB and AC, respectively, of each cable.

Given:
FB = 55 N
c = 2 m
Fc = 40 N
d = 4 m
a = 6 m
e = 4 m
b = 1.5 m
f = 3 m

Determine the magnitude and coordinate direction angles of F3 so that resultant of the three forces acts along the positive y axis and has magnitude FR.

Given:
FR = 600 lb
F1 = 180 lb
F2 = 300 lb
φ = 40 deg
θ = 30 deg

Determine the magnitude and coordinate direction angles of F3 so that resultant of the three forces is zero.

Given:
F1 = 180 lb
F2 = 300 lb
φ = 40 deg
θ = 30 deg

Resolve the force F into two components, one acting parallel and the other acting perpendicular to the u axis.

Given:
F = 600 lb
θ1 = 60 deg
θ2 = 20 deg

The force F has a magnitude F and acts at the midpoint C of the thin rod. Express the force as a Cartesian vector.
Given:
F = 80 lb
a = 2 ft
b = 3 ft
c = 6 ft

Determine the magnitude and direction of the resultant FR = F1 + F2 + F3 of the three forces by first finding the resultant F' = F1 + F3 and then forming FR = F' + F2. Specify its direction measured counterclockwise from the positive x axis.

Given:
F1 = 80 N
F2 = 75 N
F3 = 50 N
θ1 = 30 deg
θ2 = 30 deg
θ3 = 45 deg

The leg is held in position by the quadriceps AB, which is attached to the pelvis at A. If the force exerted on this muscle by the pelvis is F, in the direction shown, determine the stabilizing g force component acting along the positive y axis and the supporting force component acting along the negative x axis.

Given:
F = 85 N
θ1 = 55 deg
θ2 = 45 deg

Determine the magnitudes of the projected components of the force F in the direction of the cables AB and AC.
Given:
F = (60 12 -40) N
a = 3 m
b = 1.5 m
c = 1 m
d = 0.75 m
e = 1 m

Determine the magnitude and coordinate direction angles of F3 so that resultant of the three forces is zero.

Given:
F1 = 180 lb
φ = 40 deg
F2 = 300 lb
θ = 30 deg

Determine the angles θ and φ so that the resultant force is directed along the positive x axis and has magnitude FR.

Given:
F1 = 30 lb
F2 = 30 lb
FR = 20 l

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

Given:
F1 = 300 lb
F2 = 200 lb
θ1 = 40 deg
θ2 = 100 deg

Determine the magnitudes of F1 and F2 so that the particle is in equilibrium.

Given:
F = 500 N
θ1 = 45 deg
θ2 = 30deg

Determine the magnitude and direction θ of F so that the particle is in equilibrium.

Units Used: kN = 103 N
Given:
F1 = 7kN
F2 = 3kN
c = 4
d = 3

Determine the magnitude of F and the orientation θ of the force F3 so that the particle is in equilibrium.

Given:
F1 = 700 N
F2 = 450 N
F3 = 750 N
θ1 = 15 deg
θ2 = 30 deg

Determine the magnitude and angle θ of F so that the particle is in equilibrium. Units Used: kN = 103 N

Given:
F1 = 4.5kN
F2 = 7.5kN
F3 = 2.25kN
α = 60 deg
φ = 30 deg

The members of a truss are connected to the gusset plate. If the forces are concurrent at point O, determine the magnitudes of F and T for equilibrium. Units Used:

kN = 103 N
Given:
F1 = 8kN
F2 = 5kN
θ1 = 45 deg
θ = 30 deg

The gusset plate is subjected to the forces of four members. Determine the force in member B and its proper orientation θ for equilibrium. The forces are concurrent at point O. Units Used: kN = 103 N

Given:
F = 12kN
F1 = 8kN
F2 = 5kN
θ1 = 45 deg

Determine the maximum weight of the engine that can be supported without exceeding a tension of T1 in chain AB and T2 in chain AC.

Given:
θ = 30 deg
T1 = 450 lb
T2 = 480 lb

The engine of mass M is suspended from a vertical chain at A. A second chain is wrapped around the engine and held in position by the spreader bar BC. Determine the compressive force acting along the axis of the bar and the tension forces in segments BA and CA of the chain.

Units Used: kN = 103 N
Given:
M = 200 kg
θ1 = 55 deg
g = 9.81 m/s2

Cords AB and AC can each sustain a maximum tension T. If the drum has weight W, determine the smallest angle θ at which they can be attached to the drum.

Given:
T = 800 lb
W = 900 lb

The crate of weight W is hoisted using the ropes AB and AC. Each rope can withstand a maximum tension T before it breaks. If AB always remains horizontal, determine the smallest angle θ to which the crate can be hoisted.

Given:
W = 500 lb
T = 2500 lb

Two electrically charged pith balls, each having mass M, are suspended from light threads of equal length. Determine the resultant horizontal force of repulsion, F, acting on each ball if the measured distance between them is r.
Given:
M = 0.2 gm
r = 200 mm
l = 150 mm
d = 50 mm

The towing pendant AB is subjected to the force F which is developed from a tugboat. Determine the force that is in each of the bridles, BC and BD, if the ship is moving forward with constant velocity.

Units Used:
kN = 103 N
Given:
F = 50kN
θ1 = 20 deg
θ2 = 30 deg

Determine the stretch in each spring for equilibrium of the block of mass M. The springs are shown in the equilibrium position.
Given:
M = 2 kg
a = 3 m
b = 3 m
c = 4 m
kAB = 30 N/m
kAC = 20 N/m
g = 9.81 m/s2

The unstretched length of spring AB is δ. If the block is held in the equilibrium position shown, determine the mass of the block at D.

Given:
δ = 2 m
a = 3 m
b = 3 m
c = 4 m
kAB = 30 N/m
kAC = 20 N/m
g = 9.81 m/s2

The springs AB and BC have stiffness k and unstretched lengths l/2. Determine the horizontal force F applied to the cord which is attached to the small pulley B so that the displacement of the pulley from the wall is d.
Given:
l = 6 m
k = 500 N/m
d = 1.5 m

The springs AB and BC have stiffness k and an unstretched length of l. Determine the displacement d of the cord from the wall when a force F is applied to the cord.

Given:
l = 6 m
k = 500 N/m
F = 175 N

Determine the force in each cable and the force F needed to hold the lamp of mass M in the position shown. Hint: First analyze the equilibrium at B; then, using the result for the force in BC, analyze the equilibrium at C.

Given:
M = 4 kg
θ1 = 30 deg
θ2 = 60 deg
θ3 = 30 deg

The motor at B winds up the cord attached to the crate of weight W with a constant speed. Determine the force in cord CD supporting the pulley and the angle θ for equilibrium. Neglect the size of the pulley at C.

Given:
W = 65 lb c = 12 d = 5

The cords BCA and CD can each support a maximum load T. Determine the maximum weight of the crate that can be hoisted at constant velocity, and the angle θ for equilibrium.

Given:
T = 100 lb
c = 12
d = 5
The maximum will occur in CD rather than in BCA.

The sack has weight W and is supported by the six cords tied together as shown. Determine the tension in each cord and the angle θ for equilibrium. Cord BC is horizontal.

Given:
W = 15 lb
θ1 = 30 deg
θ2 = 45 deg
θ3 = 60 deg

Each cord can sustain a maximum tension T. Determine the largest weight of the sack that can be supported. Also, determine θ of cord DC for equilibrium.

Given:
T = 200 lb
θ1 = 30 deg
θ2 = 45 deg
θ3 = 60 deg

The block has weight W and is being hoisted at uniform velocity. Determine the angle θ for equilibrium and the required force in each cord.

Given:
W = 20 lb
φ = 30 deg

Determine the maximum weight W of the block that can be suspended in the position shown if each cord can support a maximum tension T. Also, what is the angle θ for equilibrium?

Given:
T = 80 lb
φ = 30 deg
The maximum load will occur in cord AB.

Two spheres A and B have an equal mass M and are electro statically charged such that the repulsive force acting between them has magnitude F and is directed along line AB. Determine the angle θ, the tension in cords AC and BC, and the mass M of each sphere.
Unit used:
mN 10−3 = N

Given:
F = 20 mN
g = 9.81 m/s2
θ1 = 30 deg
θ2 = 30 deg

Blocks D and F weigh W1 each and block E weighs W2. Determine the sag s for equilibrium. Neglect the size of the pulleys.
Given:
W1 = 5 lb
W2 = 8 lb
a = 4 ft

If blocks D and F each have weight W1, determine the weight of block E if the sag is s. Neglect the size of the pulleys.
Given:
W1 = 5 lb
s = 3 ft
a = 4 ft

The block of mass M is supported by two springs having the stiffness shown. Determine the unstretched length of each spring.
Units Used:
kN = 103 N
Given:
M = 30 kg
l1 = 0.6 m
l2 = 0.4 m
l3 = 0.5 m
kAC = 15 kN/m
kAB = 1.2kN/m
g = 9.81 m/s2

Three blocks are supported using the cords and two pulleys. If they have weights of WA = WC = W, WB = kW, determine the angle θ for equilibrium.

Given:
k = 0.25

A continuous cable of total length l is wrapped around the small pulleys at A, B, C, and D. If each spring is stretched a distance b, determine the mass M of each block. Neglect the weight of the pulleys and cords. The springs are unstretched when d = l/2.
Given:
l = 4 m
k = 500 N/m
b = 300 mm
g = 9.81m/s2

Prove Lami's theorem, which states that if three concurrent forces are in equilibrium, each is proportional to the sine of the angle of the other two; that is, P/sin α = Q/sin β = R/sin γ.

A vertical force P is applied to the ends of cord AB of length a and spring AC.
If the spring has an unstretched length δ, determine the angle θ for equilibrium.

Given:
P = 10 lb
δ = 2 ft
k = 15 lb/ ft
a = 2 ft
b = 2 ft

Determine the unstretched length δ of spring AC if a force P causes the angle θ for equilibrium. Cord AB has length a.

Given:
P = 80 lb
θ = 60 deg
k = 50 lb/ft
a = 2 ft
b = 2 ft

The flowerpot of mass M is suspended from three wires and supported by the hooks at B and C. Determine the tension in AB and AC for equilibrium.
Given:
M = 20 kg
l1 = 3.5 m
l2 = 2 m
l3 = 4 m
l4 = 0.5 m
g = 9.81m/s2

A car is to be towed using the rope arrangement shown. The towing force required is P. Determine the minimum length l of rope AB so that the tension in either rope AB or AC does not exceed T. Hint: Use the equilibrium condition at point A to determine the required angle θ for attachment, then determine l using trigonometry applied to triangle ABC.

Given:
P = 600 lb
T = 750 lb
φ = 30 deg
d = 4 ft

Determine the mass of each of the two cylinders if they cause a sag of distance d when suspended from the rings at A and B. Note that s = 0 when the cylinders are removed.
Given:
d = 0.5 m
l1 = 1.5 m
l2 = 2 m
l3 = 1 m
k = 100 N/m
g = 9.81m/s2

The sling BAC is used to lift the load W with constant velocity. Determine the force in the sling and plot its value T (ordinate) as a function of its orientation θ, where 0 ≤ θ ≤ 90Â°.

The lamp fixture has weight W and is suspended from two springs, each having unstretched length L and stiffness k. Determine the angle θ for equilibrium.
Units Used:
kN = 103 N

Given:
W = 10 lb
L = 4 ft
k = 5 lb/ft
a = 4 ft

The uniform tank of weight W is suspended by means of a cable, of length l, which is attached to the sides of the tank and passes over the small pulley located at O. If the cable can be attached at either points A and B, or C and D, determine which attachment produces the least amount of tension in the cable. What is this tension?
Given:
W = 200 lb
l = 6 ft
a = 1 ft
b = 2 ft
c = b
d = 2a

A sphere of mass ms rests on the smooth parabolic surface. Determine the normal force it exerts on the surface and the mass mB of block B needed to hold it in the equilibrium position shown.

Given:
ms = 4 kg
a = 0.4 m
b = 0.4 m
θ = 60 deg
g = 9.81 m/s2

The pipe of mass M is supported at A by a system of five cords. Determine the force in each cord for equilibrium.

Given:
M = 30 kg c = 3
g = 9.81 m/s2 d = 4
θ = 60 deg

The joint of a space frame is subjected to four forces. Strut OA lies in the x-y plane and strut OB lies in the y-z plane. Determine the forces acting in each of the three struts required for equilibrium.
Units Used:
kN = 103 N

Given:
F = 2kN
θ1 = 45 deg
θ2 = 40 deg

Determine the magnitudes of F1, F2, and F3 for equilibrium of the particle.
Units Used:
kN = 103 N

Given:
F4 = 800 N
α = 60 deg
β = 30 deg
γ = 30 deg
c = 3
d = 4

Determine the magnitudes of F1, F2, and F3 for equilibrium of the particle.
Units Used:
kN = 1000 N

Given:
F4 = 8.5kN
F5 = 2.8kN
α = 15 deg
β = 30 deg
c = 7
d = 24

Determine the magnitudes of F1, F2 and F3 for equilibrium of the particle F = {- 9i - 8j - 5k}.
Units Used:
kN = 103 N

Given:
F = (-9 -8 -5) kN
a = 4 m
b = 2 m
c = 4 m
θ1 = 30 deg
θ2 = 60 deg
θ3 = 135 deg
θ4 = 60 deg
θ5 = 60 deg

The three cables are used to support the lamp of weight W. Determine the force developed in each cable for equilibrium.
Units Used:
kN = 103 N
Given:
W = 800 N b = 4 m
a = 4 m c = 2 m

Determine the force in each cable needed to support the load W.
Given:
a = 8 ft
b = 6 ft
c = 2 ft
d = 2 ft
e = 6 ft
W = 500 lb

Determine the stretch in each of the two springs required to hold the crate of mass mc in the equilibrium position shown. Each spring has an unstretched length δ and a stiffness k.

Given:
mc = 20 kg
δ = 2 m
k = 300N/m
a = 4 m
b = 6 m
c = 12 m

If the bucket and its contents have total weight W, determine the force in the supporting cables
DA, DB, and DC
Given:
W = 20 lb
a = 3 ft
b = 4.5 ft
c = 2.5 ft
d = 3 ft
e = 1.5 ft
f = 1.5 ft

The crate which of weight F is to be hoisted with constant velocity from the hold of a ship using the cable arrangement shown. Determine the tension in each of the three cables for equilibrium. Units Used: kN = 103 N
Given:
F = 2.5kN
a = 3 m
b = 1 m
c = 0.75 m
d = 1 m
e = 1.5 m
f = 3 m

The lamp has mass ml and is supported by pole AO and cables AB and AC. If the force in the pole acts along its axis, determine the forces in AO, AB, and AC for equilibrium.
Given:
ml = 15 kg d = 1.5 m
a = 6 m e = 4 m
b = 1.5 m f = 1.5 m
c = 2 m g 9.81 m/s2

Cables AB and AC can sustain a maximum tension Tmax, and the pole can support a maximum compression Pmax. Determine the maximum weight of the lamp that can be supported in the position shown. The force in the pole acts along the axis of the pole.
Given:
Tmax = 500 N c = 2 m
Pmax = 300 N d = 1.5 m
a = 6 m e = 4 m
b = 1.5 m f = 1.5 m

Determine the tension in cables AB, AC, and AD, required to hold the crate of weight W in equilibrium.
Given:
W = 60 lb
a = 6 ft
b = 12 ft
c = 8 ft
d = 9 ft
e = 4 ft
f = 6 ft

The bucket has weight W. Determine the tension developed in each cord for equilibrium.
Given:
W = 20 lb
a = 2 ft
b = 2 ft
c = 8 ft
d = 7 ft
e = 3 ft
f = a

The mast OA is supported by three cables. If cable AB is subjected to tension T, determine the tension in cables AC and AD and the vertical force F which the mast exerts along its axis on the collar at A.
Given:
T = 500 N
a = 6 m
b = 3 m
c = 6 m
d = 3 m
e = 2 m
f = 1.5 m
g = 2 m

The ends of the three cables are attached to a ring at A and to the edge of the uniform plate of mass M. Determine the tension in each of the cables for equilibrium.
Given:
M = 150 kg e = 4 m
a = 2 m f = 6 m
b = 10 m g = 6 m
c = 12 m h = 6 m
d = 2 m i = 2 m
Gravity = 9.81 m/s2

The ends of the three cables are attached to a ring at A and to the edge of the uniform plate. Determine the largest mass the plate can have if each cable can support a maximum tension of T.
kN = 103 N
Given:
T = 15 kN e = 4 m
a = 2 m f = 6 m
b = 10 m g = 6 m
c = 12 m h = 6 m
d = 2 m i = 2 m
Gravity = 9.81 m/s2

The crate of weight W is suspended from the cable system shown. Determine the force in each segment of the cable, i.e., AB, AC, CD, CE, and CF. Hint: First analyze the equilibrium of point A, then using the result for AC, analyze the equilibrium of point C.
Units Used:
kip = 1000 lb

Given:
W = 500 lb
a = 10 ft
b = 24 ft
c = 24 ft
d = 7 ft
e = 7 ft
θ1 = 20 deg
θ2 = 35 deg

The chandelier of weight W is supported by three wires as shown. Determine the force in each wire for equilibrium.
Given:
W = 80 lb
r = 1 ft
h = 2.4 ft

If each wire can sustain a maximum tension Tmax before it fails, determine the greatest weight of the chandelier the wires will support in the position shown.
Given:
Tmax = 120 lb
r = 1 ft
h = 2.4 ft

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