A 9-m steel boom has a mass of 600 kg and its center of mass G is located at 6 meters from the x-y plane.
It is supported by a ball-and socket joint at point A and three cables (BF, CD, and HE). Calculate the
magnitude of the tensions in the three cables and the reaction force at point A. Note: Cables BF and BH are
directly connected to sleeve B, which is welded to the boom.
G
C
1m
I
H
2m
Figure 1
SOLUTION STRATEGY:
1) Set up the FBD of the 2000 kg mass and calculated the tension in cable FB (Drop Down, Enter
Values, or Drag and Drop)
2) Set up the FBD of bar AC (Drop Down, Enter Values, or Drag and Drop).
3) Calculate the weight of the bar.
4) Find vectors R CD , R HE , & R BF along with their corresponding unit vectors nCD , n HE , & n BF
5) Identify the vector forms of the unknown tensions TCD , THE, and TBF .
F = 0
Use  M = 0
6) Use
7)
8) Put the 6 equations and 6 unknowns in matrix form.
9) Solve for the reaction force at point A and the tensions in the three cables.
1
Link to Problem Statement (New Window)
(http://www.randjanimations.com/wiley_flash_development/meriam_voiceovers/statics/EQIP/
p3_72/p3_72_main_problem.html)
LEARNING OUTCOMES:
1) Create a FBD of a 3-D structure with multiple applied forces and tensions.
2) Calculate displacement vectors in 3-D.
3) Determine unit vectors, which define the direction of a force in 3-D.
 F = 0 ) in 3-D.
Pick the most efficient point for  M = 0 .
Use the vector approach to calculate  M = 0 in 3-D.
4) Apply Newton’s 2nd Law (
5)
6)
7) Set up and solve for the forces, which are defined by a set of linear equations with multiple
equations and unknowns via matrix methods.
2
Part 1
Enter the values of the tension TBI and the weight W into the FBD below.
TBI = 19,620 (N)
ĵ
2000
kg
I
î
W = 19,620 (N)
Figure 2
Part 2
Drag and drop the forces and enter the coordinates of points A through H into the FBD, below.
Y (m)
ENTER
COORDINATES
BELOW
(X , Y , Z)
A (0,0,0)
B (0,0,5)
C (1,0,9)
D (6,2,0)
E (-2,6,0)
F (-3,0,0)
G (0,0,6)
H (-2,0,9)
ĵ
E
D
î
X (m)
F___
AZ
k̂
A
F
___
AX
T
___
BF
F
T
CD
___
T
HE
___
B
G
F
___
AY
C
T
___
BI
___
W
Figure 3
3
Z (m)
After Parts 1-2 are Correct. Link to Parts 1-2 (New Window)
(http://www.randjanimations.com/wiley_flash_development/meriam_voiceovers/statics/EQIP/p3_72/
p3_72_P1_P2.html)
Part 3
Calculate the weight of the bar. W = 5,886 (Newtons).
Part 4
Calculate the values of the vectors R CD , R HE , and R BF . (These are in the same direction as the
corresponding forces.) Tutorial *
R CD  5 i + 2 j + -9 k (Meters)
R HE  0 i + 6 j + -9 k (Meters)
R BF  -3i + 0 j + -5 k (Meters)
Part 5
Calculate the magnitude of the vectors R CD , R HE , and R BF .Tutorial*
| R CD |  110 (Meters)
| R HE |  117 (Meters)
| R BF |  34 (Meters)
Part 6
Calculate the unit vectors, which define the directions of forces TCD , THE, and TBF . Tutorial**
1
 5 i + 2 j + -9 k 
110
1
n HE 
 0 i + 6 j + -9 k 
117
1
n BF 
 -3 i + 0 j + -5 k 
34
nCD 
After Parts 3-6 are Correct. Link to Parts 3-6 (New Window)
(http://www.randjanimations.com/wiley_flash_development/meriam_voiceovers/statics/EQIP/p3_72/
p3_72_P3_P6.html)
4
Part 7
Therefore, the force vectors
TCD , THE, and TBF can be expressed as: (Hint: These values have already been
calculated in Part 6.) Tutorial **
TCD  TCDnCD  TCD
THE  THEnCD  THE
TBF  TBFnCD  TBF
1
 5 i + 2 j + -9 k  (Newtons)
110
1
 0 i + 6 j + -9 k  (Newtons)
117
1
 -3 i + 0 j + -5 k  (Newtons)
34
Part 8
Use Newton’s 2nd Law
 F = 0 in order to obtain 3 equations. Enter the integers into the corresponding
spot in each equation. (Hint: Combine parts 1, 3, and 7.)
5
TCD +
110
2
j:
TCD +
110
-9
k:
TCD +
110
i:
0
THE +
117
6
THE +
117
-9
THE +
117
-3
TBF  1FAX  0FAY  0FA Z  0 = 0......(1)
34
0
TBF  0FAX  1FAY  0FA Z  -25,506 = 0......(2)
34
-5
TBF  0FAX  0FAY  1FA Z  0 = 0......(3)
34
Part 9
What is the most efficient point for
M = 0 ? A
NON-HTML** Link to why A is the best point.: Correct: GREAT! Point A eliminates reaction
forces FAX, FAY, and FAZ…Incorrect: TRY AGAIN: Pick the point that eliminates the most unknown
reaction forces)
After Parts 7-9 are Correct. Link to Parts 7-9 (New Window)
(http://www.randjanimations.com/wiley_flash_development/meriam_voiceovers/statics/EQIP/p3_72/
p3_72_P7_P9.html)
5
Part 10
Use
M
A
=  R X F for the moment about point A resulting from the forces at points G, B, C, and H
and enter the integers in the cross products below: Tutorial ***
Point G (the weight of the boom):
0 i + 0 j + 6 k  X0 i + -5,886 j + 0 k  = 35,316 i + 0 j + 0 k  (Nm)
Point B (TBI):
0 i + 0 j + 5 k  X0 i + -19,620 j + 0 k  = 98,100 i + 0 j + 0 k  (Nm)
Point C (TCD):
1 i + 0 j + 9 k  X
1
1
TCD 5 i + 2 j + -9 k  =
TCD -18 i + 54 j + 2 k  (Nm)
110
110
Point H (THE):
-2 i + 0 j + 9 k  X
1
1
THE  0 i + 6 j + -9 k  =
THE -54 i + -18 j + -12 k  (Nm)
117
117
Point B (TBF):
0 i + 0 j + 5 k  X
1
1
TBF -3 i + 0 j + -5 k  =
TBF 0 i + -15 j + 0 k  (Nm)
34
34
Part 11
By using the information in Part 10, enter the integers into the corresponding spot in each equation below.
-18
TCD +
110
54
j:
TCD +
110
2
k:
TCD +
110
i:
-54
THE +
117
-18
THE +
117
-12
THE +
117
0
TBF  133,306 = 0......(4)
34
-15
TBF + 0 = 0......(5)
34
0
TBF  0 = 0......(6)
34
After Parts 10-11 are Correct. Link to Parts 10-11 (New Window)
http://www.randjanimations.com/wiley_flash_development/meriam_voiceovers/statics/EQIP/p3_72/p
3_72_P10_P11.html)
6
Part 12
Put equations 1-6 in matrix form Ax = b . Enter the integers in the proper matrix and column positions.
The rows of the matrix A and column vector b must follow the order of equations 1-6. TUTORIAL ****

1


0

0


0


0


0


0 0
1 0
0 1
0 0
0 0
0 0
5
110
2
110
-9
110
-18
110
54
110
2
110
0
117
6
117
-9
117
-54
117
-18
117
-12
117
-3 
34 

0 
0

34   FAX  
  

-5   FAY   25,506 

0
34   FAZ  
   

0  TCD  -133,416 

34  THE  
0
  

-15   TBF  
0

34 

0 

34 
Part 13
Solve for FAX ,FAY , FAZ ,TCD ,THE , & TBF (use computer methods).
FAX = 25.7 (kN)
FAY = 10.8 (kN)
FAZ = 135.9 (kN)
TCD = 51.8 (kN)
THE = 8.9 (kN)
TBF = 98.0 (kN)
After Parts 12-13 are Correct. Link to Parts 12-13 (New Window)
(http://www.randjanimations.com/wiley_flash_development/meriam_voiceovers/statics/EQIP/p3_72/
p3_72_P12_P13.html)
7
* Hyperlink to
http://www.randjanimations.com/wiley_flash_development/meriam_voiceovers/statics/r_vector_3_d/r_vector_3
_d.html
** Hyperlink to
http://www.randjanimations.com/wiley_flash_development/meriam_voiceovers/statics/force_unit_vectors/
force_unit_vectors.html
*** Hyperlink to
http://www.randjanimations.com/Wiley_Flash_Development/meriam_voiceovers/statics/T16_3d_cross_pr
oduct/T16_3d_cross_product.html
**** Hyperlink to
http://www.randjanimations.com/wiley_flash_development/meriam_voiceovers/statics/algebraic_linear_eq
ns_matrix/algebraic_linear_eqns_matrix.html
8
Notes to programmers
1)
2)
3)
4)
5)
6)
7)
All entries are numbers, except for Part 2.
All numbers are integers, except for the final answer (Part 13), which should be accurate within 1
decimal point.
The “drag and drop” values could be a drop-down list (Part 2).
Some of the numeric entries could be already filled in and the students would be required to calculated
the remaining entries (Parts 2, 4-8, 10-12)
Links to voice-over tutorials could be added to steps that involve mathematical concepts (e.g. cross
product: Part 10, transferring a system of equations into a matrix format: Part 12, etc.).
A drop-down menu could be used for Part 9, with the possible answers of all points (A-H).
Note that the vectors are Bold, instead of “hats” or “arrow”. Also, scalars are non-bold. This is
consistent with the text.
9