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A sewer pipe is to be laid from station 10+00 to station 13+20 on a -1% grade, starting with invert elevation 326.32ft. At 10+00 Calculate invert elevations at each 50ft station along line.
Determine the magnitude of the resultant force FR = F1 + F2 and its direction, measured counterclockwise from the positive x axis.

Given:
F1 = 600 N
F2 = 800 N
F3 = 450 N
α = 45 deg
β = 60 deg
γ = 75 deg


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

Given:
F1 = 80 lb
F2 = 60 lb
θ = 120 deg

Its direction measured counterclockwise from the positive x axis
Determine the magnitude of the resultant force FR = F1 + F2 and its direction, measured counterclockwise from the positive x axis.

Given:
F1 = 250 lb
F2 = 375 lb
θ = 30 deg
φ = 45 deg


The resultant force FR = F1 + F2 and its
Determine the magnitude of the resultant force FR = F1 + F2 and its direction, measured counterclockwise from the positive u axis.

Given:
F1 = 300 N
F2 = 500 N
α = 30 deg
β = 45 deg
γ = 70 deg

Measured counterclockwise from the positive u axis
Resolve the force F1 into components acting along the u and v axes and determine the magnitudes of the components.

Given:
F1 = 300 N
α = 30 deg
F2 = 500 N
β = 45 deg
γ = 70 deg

Resolve the force F1 into components acting
Resolve the force F2 into components acting along the u and v axes and determine the magnitudes of the components.

Given:
F1 = 300 N
F2 = 500 N
α = 30 deg
β = 45 deg
γ = 70 deg

the magnitudes of the components
Determine the magnitude of the resultant force FR = F1 + F2 and its direction measured counterclockwise from the positive u axis.

Given:
F1 = 25 lb
F2 = 50 lb
θ1 = 30 deg
θ2 = 30 deg
θ3 = 45 deg

force FR = F1 + F2 and its direction
Resolve the force F1 into components acting along the u and v axes and determine the components.

Given:
F1 = 25 lb
F2 = 50 lb
θ1 = 30 deg
θ2 = 30 deg
θ3 = 45 deg

acting along the u and v axes and determine
Resolve the force F2 into components acting along the u and v axes and determine the components.

Given:
F1 = 25 lb
F2 = 50 lb
θ1 = 30 deg
θ2 = 30 deg
θ3 = 45 deg

Resolve the force F2 into components
Determine the components of the F force acting along the u and v axes.

Given:
θ1 = 70 deg
θ2 = 45 deg
θ3 = 60 deg
F = 250 N

Determine the components of the F force acting along the u and v
The force F acts on the gear tooth. Resolve this force into two components acting along the lines aa and bb.

Given:
F = 20 lb
θ1 = 80 deg
θ2 = 60 deg

The force F acts on the gear tooth.
The component of force F acting along line aa is required to be Fa. Determine the magnitude of F and its component along line bb.

Given:
Fa = 30 lb
θ1 = 80 deg
θ2 = 60 deg

Determine the magnitude of F and its component along line bb.
A resultant force F is necessary to hold the balloon in place. Resolve this force into components along the tether lines AB and AC, and compute the magnitude of each component.

Given:
F = 350 lb
θ1 = 30 deg
θ2 = 40 deg

Force F is necessary to hold the balloon in place
The post is to be pulled out of the ground using two ropes A and B. Rope A is subjected to force F1 and is directed at angle θ1 from the horizontal. If the resultant force acting on the post is to be FR, vertically upward, determine the force T in rope B and the corresponding angle θ.

Given:
FR = 1200 lb
F1 = 600 lb
θ1 = 60 deg

The post is to be pulled out of the ground
Resolve the force F1 into components acting along the u and v axes and determine the magnitudes of the components.

Given:
F1 = 250 N
F2 = 150 N
θ1 = 30 deg
θ2 = 30 deg
θ3 = 105 deg

F1 into components acting along the u and v
Resolve the force F2 into components acting along the u and v axes and determine the magnitudes of the components.

Given:
F1 = 250 N
F2 = 150 N
θ1 = 30 deg
θ2 = 30 deg
θ3 = 105 deg

The magnitudes of the components along the u and v axes
Determine the magnitude and direction of the resultant force FR. Express the result in terms of the magnitudes of the components F1 and F2 and the angle φ.

Determine the magnitude and direction of the resultant
If the tension in the cable is F1, determine the magnitude and direction of the resultant force acting on the pulley. This angle defines the same angle θ of line AB on the tailboard block.

Given:
F1 = 400 N
θ1 = 30 deg

If the tension in the cable is F1, determine the
The riveted bracket supports two forces. Determine the angle θ so that the resultant force is directed along the negative x axis. What is the magnitude of this resultant force?

Given:
F1 = 60 lb
F2 = 70 lb
θ1 = 30 deg

The riveted bracket supports two forces
The plate is subjected to the forces acting on members A and B as shown. Determine the magnitude of the resultant of these forces and its direction measured clockwise from the positive x axis.

Given:
FA = 400 lb
FB = 500 lb
θ1 = 30 deg
θ = 60 deg

The plate is subjected to the forces acting on
Determine the angle θ for connecting member B to the plate so that the resultant of FA and FB is directed along the positive x axis. What is the magnitude of the resultant force?

Given:
FA = 400 lb
FB = 500 lb
θ1 = 30 deg

Determine the angle θ for connecting member B
Determine the magnitude and direction of the resultant FR = F1 + F2 + F3 of the three forces by first finding the resultant F' = F1 + F2 and then forming FR = F' + F3.

Given:
F1 = 30 N
F2 = 20 N
F3 = 50 N
θ = 20 deg
c = 3
d = 4

Three forces by first finding the resultant F
Determine the magnitude and direction of the resultant FR = F1 + F2 + F3 of the three forces by first finding the resultant F' = F2 + F3 and then forming FR = F' + F1.

Given:
F1 = 30 N
F2 = 20 N
F3 = 50 N
θ = 20 deg
c = 3
d = 4




Resolve the force F into components acting along
(a) The x and y axes, and
(b) The x and y' axes.
Given:

F = 50 lb
α = 65 deg
β = 45 deg
γ = 30 deg




The boat is to be pulled onto the shore using two ropes. Determine the magnitudes of forces T and P acting in each rope in order to develop a resultant force F1, directed along the keel axis aa as shown.
Given:

θ = 40 deg
θ1 = 30 deg
F1 = 80 lb




The boat is to be pulled onto the shore using two ropes. If the resultant force is to be F1, directed along the keel aa as shown, determine the magnitudes of forces T and P acting in each rope and the angle θ of P so that the magnitude of P is a minimum. T acts at θ from the keel as shown.
Given:

θ1 = 30 deg
F1 = 80 lb




The beam is to be hoisted using two chains. Determine the magnitudes of forces FA and FB acting on each chain in order to develop a resultant force T directed along the positive y axis.
Given:

T = 600 N
θ1 = 30 deg
θ = 45 deg




The beam is to be hoisted using two chains. If the resultant force is to be F, directed along the positive y axis, determine the magnitudes of forces FA and FB acting on each chain and the orientation θ of FB so that the magnitude of FB is a minimum.

Given:
F = 600 N
θ1 = 30 deg




Three chains act on the bracket such that they create a resultant force having magnitude FR. If two of the chains are subjected to known forces, as shown, determine the orientation θ 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.
Given:

FR = 500 lb
F1 = 200 lb
F2 = 300 lb
φ = 30 deg




Three cables pull on the pipe such that they create a resultant force having magnitude FR. If two of the cables are subjected to known forces, as shown in the figure, determine the direction θ of the third cable so that the magnitude of force F in this cable 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.
Given:

FR = 900 lb
F1 = 600 lb
F2 = 400 lb
α = 45 deg
β = 30 deg




Determine the x and y components of the force F.S
Given:
F = 800 lb
α = 60 deg
β = 40 deg




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

F1 = 70 N
F2 = 50 N
F3 = 65 N
θ = 30 deg
φ = 45 deg




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

F1 = 50 lb
F2 = 35 lb
α = 120 deg
β = 25 deg




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

F1 = 850 N
F2 = 625 N
F3 = 750 N
θ = 45 deg
φ = 30 deg
c = 3
d = 4




Three forces act on the bracket. Determine the magnitude and direction θ of F1 so that the resultant force is directed along the positive x' axis and has a magnitude of FR. Units Used: kN = 103 N
Given:

FR = 1kN
F2 = 450 N
F3 = 200 N
α = 45 deg
β = 30 deg




Determine the magnitude and direction, measured counterclockwise from the x' axis, of the resultant force of the three forces acting on the bracket.
Given:
F1 = 300 N
F2 = 450 N
F3 = 200 N
α = 45 deg
β = 30 deg
θ = 20 deg




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

F1 = 800 N
F2 = 600 N
θ = 40 deg
c = 12
d = 5




Determine the magnitude of the resultant force and its direction, measured counterclockwise from the positive x axis.
Units Used: kN = 103 N
Given:

F1 = 30kN
F2 = 26kN
θ = 30 deg
c = 5
d = 12




Determine the magnitude of the resultant force and its direction measured counterclockwise from the positive x axis.
Given:
F1 = 60 lb
F2 = 70 lb
F3 = 50 lb
θ1 = 60 deg
θ2 = 45 deg
c = 1
d = 1




Determine the magnitude of the resultant force FR = F1 + F2 and its direction, measured counterclockwise from the positive x axis by summing the rectangular or x, y components of the forces to obtain the resultant force.
Given:

F1 = 600 N
F2 = 800 N
F3 = 450 N
θ1 = 60 deg
θ2 = 45 deg
θ3 = 75 deg




Determine the magnitude and direction of the resultant FR = F1 + F2 + F3 of the three forces by summing the rectangular or x, y components of the forces to obtain the resultant force.
Given:

F1 = 30 N
F2 = 20 N
F3 = 50 N
θ = 20 deg
c = 3
d = 4




Determine the magnitude and orientation, measured counterclockwise from the positive y axis, of the resultant force acting on the bracket.
Given:

FA = 700 N
FB = 600 N
θ = 20 deg
φ = 30 deg




Determine the magnitude and direction, measured counterclockwise from the positive x' axis, of the resultant force of the three forces acting on the bracket.

Given:
F1 = 300 N
F2 = 200 N
F3 = 180 N
θ = 10 deg
θ1 = 60 deg
c = 5
d = 12




Determine the x and y components of F1 and F2.
Given:
F1 = 200 N
F2 = 150 N
θ = 45 deg
φ = 30 deg




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

Given:
F1 = 200 N
F2 = 150 N
θ = 45 deg
φ = 30 deg




Determine the x and y components of each force acting on the gusset plate of the bridge truss.
Given:
F1 = 200 lb c = 3
F2 = 400 lb d = 4
F3 = 300 lb e = 3
F4 = 300 lb f = 4




Determine the magnitude of the resultant force and its direction measured clockwise from the positive x axis. Units Used: kN = 103 N

Given:
F1 = 20 kN
F2 = 40 kN
F3 = 50 kN
θ = 60 deg
c = 1
d = 1
e = 3
f = 4




Three forces act on the bracket. Determine the magnitude and direction θ of F1 so that the resultant force is directed along the positive x' axis and has magnitude FR.

Given:
F2 = 200 N
F3 = 180 N
θ1 = 60 deg
FR = 800 N
c = 5
d = 12




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

Given:
F1 = 300 N
F2 = 200 N
F3 = 180 N
θ1 = 60 deg
θ = 10 deg
c = 5
d = 12




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

Given:
F1 = 150 lb θ = 60 deg
F2 = 275 lb c = 4
F3 = 75 lb d = 3




Determine the magnitude of force F so that the resultant FR of the three forces is as small as possible. What is the minimum magnitude of FR?
Units Used: kN = 1000 N

Given:
F1 = 5 kN
F2 = 4 kN
θ = 30 deg




Express each of the three forces acting on the bracket in Cartesian vector form with respect to the x and y axes. Determine the magnitude and direction θ of F1 so that the resultant force is directed along the positive x' axis and has magnitude FR. Units Used:
KN = 1000 N
Given:
FR = 600 N
F2 = 350 N
F3 = 100 N
φ = 30 deg




The three concurrent forces acting on the post produce a resultant force FR = 0. If F2 = (1/2)F1, and F1 is to be 90° from F2 as shown, determine the required magnitude F3 expressed in terms of F1 and the angle θ.




Three forces act on the bracket. Determine the magnitude and orientation θ of F2 so that the resultant force is directed along the positive u axis and has magnitude FR.

Given:
FR = 50 lb
F1 = 80 lb
F3 = 52 lb
φ = 25 deg
c = 12
d = 5




Determine the magnitude and orientation, measured clockwise from the positive x axis, of the resultant force of the three forces acting on the bracket.

Given:
F1 = 80 lb
F2 = 150 lb
F3 = 52lb
θ = 55 deg
φ = 25 deg
c = 12 m
d = 5 m




Three forces act on the ring. Determine the range of values for the magnitude of P so that the magnitude of the resultant force does not exceed F. Force P is always directed to the right.
Units Used: kN = 103 N

Given:
F = 2500 N
F1 = 1500 N
F2 = 600 N
θ1 = 60 deg
θ2 = 45 deg




Determine the magnitude and coordinate direction angles of F1 and F2. Sketch each force on an x, y, z reference. Given:




Express each force in Cartesian vector form.
Units Used: kN = 103 N

Given:
F1 = 5kN
F2 = 2kN
θ1 = 60 deg
θ2 = 60 deg
θ3 = 45 deg




Determine the magnitude and coordinate direction angles of the force F acting on the stake.

Given:
Fh = 40N
θ = 70deg
c = 3
d = 4




Express each force in Cartesian vector form.

Given:
F1 = 400 lb
F2 = 600 lb
θ1 = 45 deg
θ2 = 60 deg
θ3 = 60 deg
θ4 = 45 deg
θ5 = 30 deg




The stock S mounted on the lathe is subjected to a force F, which is caused by the die D. Determine the coordinate direction angle β and express the force as a Cartesian vector. Given: F = 60 N α = 60 deg γ = 30 deg
Determine the magnitude and coordinate direction angles of the resultant force.

Given:
F1 = 80 lb
F2 = 130 lb
θ = 40 deg
φ = 30 deg




Specify the coordinate direction angles of F1 and F2 and express each force as a Cartesian vector.

Given:
F1 = 80 lb
F2 = 130 lb
φ = 30 deg
θ = 40 deg




The mast is subjected to the three forces shown. Determine the coordinate angles α1, β1, γ1 of F1 so that the resultant force acting on the mast is FRi.

Given:
FR = 350 N
F1 = 500 N
F2 = 200 N
F3 = 300 N




The mast is subjected to the three forces shown. Determine the coordinate angles α1, β1, γ1 of F1 so that the resultant force acting on the mast is zero.

Given:
F1 = 500 N
F2 = 200 N
F3 = 300 N




The shaft S exerts three force components on the die D. Find the magnitude and direction of the resultant force. Force F2 acts within the octant shown.

Given:
F1 = 400 N
F2 = 300 N
F3 = 200 N
α2 = 60 deg
γ2 = 60 deg
c = 3
d = 4




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.

Given:
F1 = 630 lb α = 60 deg
F2 = 250 lb β = 135 deg
c = 24 γ = 60 deg
d = 7




Determine the magnitude and coordinate direction angles of the resultant force.

Given:
F1 = 350 N α = 60 deg
F2 = 250N β = 60 deg
c = 3 γ = 45 deg
d = 4 θ = 30 deg




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

Given:
F = 600 lb
F1 = 180 lb
F2 = 300 lb
α1 = 30 deg
α2 = 40 deg




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

Given:
F1 = 180 lb α 1 = 30 deg
F2 = 300 lb α 2 = 40 deg




Specify the magnitude F3 and directions α3, β3, and γ3 of F3 so that the resultant force of the three forces is FR.
Units Used:
kN = 103 N

Given:
F1 = 12kN c = 5
F2 = 10kN d = 12
θ = 30 deg
FR = (0 9 0)kN




The pole is subjected to the force F, which has components acting along the x, y, z axes as shown. Given β and γ, determine the magnitude of the three components of F.
Units Used:
kN = 1000 N

Given:
F = 3kN
β = 30 deg
γ = 75 deg




The pole is subjected to the force F which has components Fx and Fz. Determine the magnitudes of F and Fy.
Units Used:
kN = 1000 N

Given:
Fx = 1.5kN
Fz = 1.25kN
β = 75 deg




The eye bolt is subjected to the cable force F which has a component Fx along the x axis, a component Fz along the z axis, and a coordinate direction angle β.
Determine the magnitude of F.

Given:
Fx = 60 N
Fz = −80 N
β = 80 deg




Three forces act on the hook. If the resultant force FR has a magnitude and direction as shown, determine the magnitude and the coordinate direction angles of force F3.

Given:
FR = 120 N
F1 = 80 N
F2 = 110 N
c = 3
d = 4
θ = 30 deg
φ = 45 deg




Determine the coordinate direction angles of F1 and FR.

Given:
FR = 120 N
F1 = 80 N
F2 = 110 N
c = 3
d = 4
θ = 30 deg
φ = 45 deg




The pole is subjected to the force F, which has components acting along the x, y, z axes as shown. Given the magnitude of F and the angles α and γ, determine the magnitudes of the components of F.

Given:
F = 80 N α = 60 deg γ = 45 deg




Two forces F1 and F2 act on the bolt. If the resultant force FR has magnitude FR and coordinate direction angles α and β, as shown, determine the magnitude of F2 and its coordinate direction angles.

Given:
F1 = 20 lb
FR = 50 lb
α = 110 deg
β = 80 deg




Given r1, r2, and r3, determine the magnitude and direction of r = 2r1 − r2 + 3r3. Given:




Represent the position vector r acting from point A (a, b, c) to point B (d, e, f) in Cartesian vector form. Determine its coordinate direction angles and find the distance between points A and B.

Given:
a = 3 m
b = 5 m
c = 6 m
d = 5 m
e = −2 m
f = 1 m
A position vector extends from the origin to point A (a, b, c). Determine the angles α, β, γ which the tail of the vector makes with the x, y, z axes, respectively.

Given:
a = 2 m b = 3 m c = 6 m
Express the position vector r in Cartesian vector form; then determine its magnitude and coordinate direction angles.

Given:
a = 4 m
b = 8 m
c = 3 m
d = 4 m




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

Given:
a = 8 ft
b = 2 ft
c = 5 ft
θ = 30 deg
φ = 20 deg




Determine the length of the connecting rod AB by first formulating a Cartesian position vector from A to B and then determining its magnitude.

Given:
b = 16 in
a = 5 in
α = 30 deg




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.

Given:
a = 1.2 m
b = 0.8 m
c = 0.3 m
d = 1.5 m
θ = 40 deg




The positions of point A on the building and point B on the antenna have been measured relative to the electronic distance meter (EDM) at O. Determine the distance between A and B. Hint: Formulate a position vector directed from A to B; then determine its magnitude.

Given:
a = 460 m
b = 653 m
α = 60 deg
β = 55 deg
θ = 30 deg
φ = 40 deg




Determine the lengths of cords ACB and CO. The knot at C is located midway between A and B.

Given:
a = 3 ft
b = 6 ft
c = 4 ft




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

Given:
a = 400
b = 125
θ = 25 deg




Determine the length of wires AD, BD, and CD. The ring at D is midway between A and B.
Given:
a = 0.5 m
b = 1.5 m
c = 2 m
d = 2 m
e = 0.5 m


Express force F as a Cartesian vector; then determine its coordinate direction angles.

Given:
F = 600 lb c = 3 ft
a = 1.5 ft φ = 60 deg
b = 5 ft




Express force F as a Cartesian vector; then determine its coordinate direction angles.

Given:
a = 1.5 ft
b = 5 ft
c = 3 ft
θ = 60 deg
F = 600 lb



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

Given:
F1 = 150 N
F2 = 200 N
a = 1.5 m
b = 4 m
c = 3 m
d = 2 m
e = 3 m
θ = 60 deg




The plate is suspended using the three cables which exert the forces shown. Express each force as a Cartesian vector.

Given:
FBA = 350 lb
FCA = 500 lb
FDA = 400 lb
a = 3 ft
b = 3 ft
c = 6 ft
d = 14 ft
e = 3 ft
f = 3 ft
g = 2 ft




The engine of the lightweight plane is supported by struts that are connected to the space truss that makes up the structure of the plane. The anticipated loading in two of the struts is shown. Express each of these forces as a Cartesian vector.

Given:
F1 = 400 lb
F2 = 600 lb
a = 0.5 ft
b = 0.5 ft
c = 3.0 ft
d = 2.0 ft
e = 0.5 ft
f = 3.0 ft




The window is held open by cable AB determine the length of the cable and express the force F acting at A along the cable as a Cartesian vector.

Given:
a = 300 mm
b = 500 mm
c = 150 mm
d = 250 mm
θ = 30 deg
F = 30 N




The force acting on the man, caused by his pulling on the anchor cord, is F. If the length of the cord is L, determine the coordinates A(x, y, z) of the anchor.

Given:
F = (40 20 −50) N
L = 25 m




Express each of the forces in Cartesian vector form and determine the magnitude and coordinate direction angles of the resultant force.

Given:
F1 = 80 lb c = 4 ft
F2 = 50 lb d = 2.5 ft
a = 6 ft e = 12
b = 2 ft f = 5




The cable attached to the tractor at B exerts force F on the framework. Express this force as a Cartesian vector.

Given:
F = 350 lb
a = 35 ft
b = 50 ft
θ = 20 deg




The cable OA exerts force F on point O. If the length of the cable is L, what are the coordinates (x, y, z) of point A?
Given:
F = (40 60 70)
L = 3 m


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