# STEPS Calculate the unit vector nBC and the resulting tension

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```					          A right angle pipe OAB is shown in Figure 1. Replace the 750-N
tensile force, which the cable exerts on point B by a force-couple
system at point O.

Figure 1

STEPS:

1) Calculate the unit vector nBC and the resulting tension vector TBC.
2) Calculate the unit vector nAC and the magnitude of the TBC along line OC.

LEARNING OUTCOMES:
1) Calculate a unit vector, given two (x,y,z) coordinates (3-D).
2) Calculate a force vector in the direction of a specific unit vector (3-D).
3) Calculate the magnitude of a force vector along a given direction (3-D).

1) Difficulty Level: 9/10.
Part 1

Enter the coordinates of points A, B, and C and the tension magnitude in Figure 2.

(X,Y,Z) (m)
O (0, 0, 0)                                                      j
B (1.6, -0.4, 0.7)
C (0, 0.7, 1.2)
i
k
ROC
nOC     O

TBC = 750 N
nB
C

Figure 2
Note to programmers: Points O and C are exact. Point B must be to one decimal point.

Part 2

The displacement vector in the direction of the tension TBC (RBC) is:

R BC  -1.6 i + 1.1 j + 0.5 k (m)

Note to programmers: Numerical entries must be to one decimal point..
Part 3

The magnitude of the RCB is:

R BC  4 =2(m)

Note to programmers: Numerical entries must be integers.

Part 4

So the unit vector in the direction of the tension T is:

1
n AB      1.6 i - 1.1 j - 0.5 k  (m)
2

Note to programmers: Numerical entry must be an integer.

Part 5

Therefore, the tension vector TBC is:

1                           
TBC  750   -1.6 i + 1.1 j + 0.5 k   (N  m)
2                           
Note to programmers: Numerical entry must be exact.

Part 6

So, the tension vector TBC is:

TBC  -600.0 i + 412.5 j  187.5 k (N)

Note to programmers: Numerical entries must to 1 decimal place.

Part 7

In order to calculate the couple at point O, we must define a vector from point O to a point on the
force (any displacement vector from O to the force will work). The most convenient point is
point C, because the mathematics is relatively simple. With this in mind, the displacement vector
from point O to C (ROC) is:

R OC  0 i + 0.7 j + 1.2 k (m)

Note to programmers: Numerical entries must be exact.
Part 8

In order to shift the tension TCB to point O, we must include a couple. The couple (or, in general,
the moment) at point O (MO) is the ROC X TCB, which is:


M O = R OC X TCB   0 i + 0.7 j + 1.2 k  X -600.0 i + 412.5 j  187.5 k (Nm)    
Tutorial: How to perform a 3-D Cross-Product
(http://www.randjanimations.com/Wiley_Flash_Development/meriam_voiceovers/statics/T
16_3d_cross_product/T16_3d_cross_product.html)

Note to programmers: Numerical entries must be to 1 decimal point

Part 9

By performing the calculations of the cross-product, MO becomes:

MO = ROC X TCB   -626.3 i + -720.0 j + 420.0 k  (Nm)

Note to programmers: Numerical entries must be to 1 decimal point

Part 10

So the magnitude of MO is

MO = (-626.3)2 +  -720.0 +  420.0 = 1042.0 (Nm)
2            2

Note to programmers: Numerical entries must be to 1 decimal point
Part 11

And finally, the statically equivalent force-couple system is:
TO = 750 N
j
MO = 1042.0 Nm

i
k
O

Figure 3
Note to programmers: Variable TO is an integer and MO must be accurate to 1 decimal
point.

Notes to programmers
Font Coding:

1) Blue underline indicates an INPUT box.
2) Red indicates notes.
3) Black indicates static text.

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