While riding a zip line from vertical pole AO to vertical pole DG, a 200 lb man comes to
equilibrium at point P which is half the distance to pole DG. The zip line APD is 101 ft
long, and the vertical pole AO is 30 ft high and supported by two cables AB and AC. The
direction angles measure from point A for cable AB are  = 110o and  = 138.6o.
A) What are the coordinates of point P?
B) What is the tension in the zip line (assume a frictionless pulley at point P)?
C) Write the forces exerted on point A by the three cables (AB, AC, and the zip
line) and the pole in Cartesian vector form (you must simplify, but you may
leave in terms of an unknown magnitude).
D) What are the tensions in cables AB and AC?
E) What is the upward force exerted by pole AO at point A?
Answers:
P (40, 22.91, ‐30) ft
Tzipline = 712.4 lb
TAB = 1107 lb; TAC = 1658 lb; Fpole = 2200 lb
The buoy is held in equilibrium by three straight anchor cables and provides an upward
force in rough seas at point O. Points A, B, and C lie on the ocean floor at the same
depth. The direction angle for cable OC measured from point O is  = 1100. The tension
in cable OA is 25 kN, and the length of cable OC is 35 m.
A) What are the direction angles  and  for cable OC as measured from point O (be
mindful from where the direction angles are measured)?
B) Write all four forces in Cartesian vector form (you must simplify, but you may
leave in terms of an unknown magnitude).
C) What are the tensions in cables OB and OC?
D) What is the upward force exerted by the buoy?
Answers:
 = 125o
FOB = 26.1 kN: FOC = 35.6 kN; Fbuoy = 57.8 kN
The 7.2’ boom OA and cables AC and AB are used to support the load W applied
downward at point A. The tension in cable AC is 10 kips, and the direction angle
(measured from A) is  = 1250. Points B and C are at the same elevation (z‐coordinate)
and lie on the x‐z plane.
A) What are the coordinates of point A?
B) What are the direction angles  and  for cable AC as measured from point
A?
C) Write the two cable forces, the weight, and the compressive force (points
toward A) acting at point A in vector form (simplify but leave in terms of
unknowns).
D) What is the tension in cable AB and the compressive force in boom OA?
E) Determine the magnitude of W.
Answers:
A (2.13, 5.86, 3.6)ft
 = 135.11o;  = 65.73o
FAB = 4.11 kips; FOA = 12.49 kips; W = 12.15 kips
The 30’ high pole AO is supported by 3 cables AE, AD, and AG such that the tension in
cable AE is 10 kips and the direction angles for cable AG (measured from A) are  =  =
700. The 20’ long cable BGC is routed around the frictionless pulley at G and lies within
the same plane as cable AG. An equilateral triangle is formed by points B, C, and G.
Points B, C, D, E, and O are at the same elevation (on the x‐y plane).
A) What is the direction angle  for cable AG as measured from point A?
B) Write the three cable forces acting at point A in vector form (simplify but leave
in terms of unknowns).
C) What are the tensions in cables AD and AG?
D) What is the tension in cable BGC?
Answers:
 = 151.07o
TAD = 5.11 kips; TAG = 7.05 kips
TBCG = 4.07 kips
The two masses are held in equilibrium on the smooth ramp by the weightless spring
and pulley system attached to bar CD. Mass B is 20 kg, and the spring constants k1 and
k2 are 500 N/m and 375 N/m, respectively. The first spring (k1) is stretched 250 mm, and
the initial length of the second spring (k2) is 500 mm. If the maximum moment about
point D that bar CD can withstand is 1165 N‐m, determine:
A)
B)
C)
the mass of A;
the normal force exerted on mass A by the frictionless surface; and
the final length of the second spring (k2).
1.5 m
0.2 m
2.5 m
C
D
k1
A
B
45o
Answers:
mA = 17.47 kg
NA = 121.2 N
Lf = 0.7 m
k2
60
o
For the system shown below, all springs are unstretched when x = 24 in. The spring
constants k1 = 10 lb/in, k2 = 5 lb/in, and k3 = k4 = 20 lb/in. The weight of block A is 100 lb
and block B is 40 lb. When a force F is applied to the rope, block B advances to the right,
block A moves vertically, the springs in series are stretched, and the springs in parallel
are compressed. If x = 30 in., determine:
A) the change in length of spring k2,
B) the force in spring k3,
C) the normal force exerted on block B by block A, and
D) the force F necessary to maintain equilibrium.
Note: Assume all surfaces are smooth and all pulleys are frictionless and
weightless,
Answers:
L2 = 4”
F3 = 69.2 lb
NA = 275 lb
F = 52.5 lb
For the system shown below, the weight of block B is 50 lb and block E is 125 lb. The
spring constants for springs k1 and k2 are 25 lb/in and 50 lb/in, respectively. If the
moment about point D is 1720 lb‐ft (clockwise), determine
A) the maximum amount that the spring k1 can be stretched;
B) the normal force exerted on block E by the ramp; and
C) the spring constant k3 if springs k2 and k3 are stretched a combined distance of
10 in.
Answers:
L = 8.2”
NE = 113.3 lb
k3 = 65.1 lb/in
The system shown below is in equilibrium and consists of two weights on a smooth
ramp, four frictionless pulleys, a spring pack, a horizontal bar ABC, and some ropes. The
spring pack is composed of two identical springs each with a spring constant of k1 = k2 =
500 lb/ft. After the system reached equilibrium, it was determined that the spring pack
stretched 3 in. and the moment generated about point A due to the forces acting on the
horizontal bar at points B and C was 3 kip‐ft.
A) Determine the weight of block X.
B) Determine the normal force exerted on block X by the ramp.
C) Determine the weight of block Y.
D) Determine the normal force exerted on block Y by the ramp.
Answers:
Wx = 7.84 kips; Nx = 7.36 kips; Wy = 2.46 kips; Ny = 2.31 kips
For the system consisting of two forces and a couple moment shown below, F1 is
perpendicular to the plane ABCD and passes through the point P (1.2, 0.4, 2) m. The
direction angles for the couple moment are = 30o and  = 70o. Force F2 lies within the
y‐z plane.
A) Express both forces and the couple moment in vector form.
B) Replace the system shown below with a single force and couple moment at
point E.
C) Can this system at point E be replaced by a single force? If so, determine the
location on the x‐y plane where the force should be located. If not,
determine the angle between the single force and couple moment at E.
D) Determine the moment about the z ‐axis due to the system shown below.
Answers:
Fr = 16i + 21.6j + 11.5k [N]; Mr = 107.2i + 1.1j + 26.2k [N‐m]
No;  = 51o
Mz‐axis = 26.2k [N‐m]
For the system of forces and moments shown below, FA and FC are equal in magnitude
but opposite in direction. The force FD has a magnitude of 300 N and is perpendicular to
the plane ABCD. The moment MB has a magnitude of 1500 N‐m and direction angles (as
measured from point B) of  = 95o and  = 80o.
A) If the moment about the y‐axis generated by the couple (consisting of FA and FC)
has a magnitude of 2900 N‐m, determine the magnitude of FA and FC.
B) Write (in vector form) the single equivalent force and couple moment that can
be placed at point E which represents the entire system of forces and moments
shown below.
C) Can this system consisting of a single force and couple moment be further
reduced to a single force? If so, determine the location on the y‐z plane where
the force should be placed. If not, determine the angle between the single
equivalent force and couple moment.
Answers:
FA = FC = 494.3 N
Fr = 240j + 180k [N]; Mr = 213.4i + 2328.8j + 990.8k [N‐m]
No;  = 14.6o
For the system shown below,
A) Write the forces and moment in Cartesian vector form;
B) Determine the moment about the x‐axis due to the 60 N force;
C) Replace the system with a single force and couple moment about the origin; and
D) Determine if this new system can be replaced by a single force. If so, locate the
force.
Note: The line of action of the 25 N‐m couple lies in the x‐y plane.
12
y
11
25 N-m
35 mm
60 N
7 mm
40 mm
40o
55 mm
50o
42 N
z
x
The 20 ft diameter, 20 ft tall tank is subjected to the two forces and the couple moment
shown below. Point C is located at coordinates (0,10,10) ft, and point B is located 10 ft
from the edge of the base of the tank. F1 = 2.5 kips, F2 = 4.2 kips, and MC = 21 kip‐ft. MC
lies in the y‐z plane.
A) Express both forces and the couple moment in vector form.
B) Replace the system shown below with a single force and couple moment at point
P.
C) Can this system be replaced by a single force? If so, determine the location on
top of the tank where the force should be located. If not, determine the angle
between the force and couple moment.