Find the moment of force F about line AB. Express in Cartesian vector form.

z
F= 2050 N
9
40
C
40 m
40 m
x
A
30 m
B
y 
Solution:
(A) Calculate the moment of force F about any point on line AB. M=r x F
(B) Find the component of that moment in the AB direction by calculating the dot product of the
moment vector and the unit vector in the AB direction.
(C) Express in Cartesian vector form.
Using point B on line AB as the reference for the moment:
1- Express force F as a Cartesian vector:
𝐹⃗ = |𝐹| 𝑢
⃗⃗𝐹
𝑢
⃗⃗𝐹 =
−40𝑗 + 9𝑘
√402 + 92
𝐹⃗ = |𝐹| 𝑢
⃗⃗𝐹 = 2050 (
=
−40𝑗 + 9𝑘
41
−40𝑗 + 9𝑘
) = −2000𝑗 + 450𝑘
41
2- Find vector r:
𝑟⃗𝑏𝑐 = −30𝑗 + 40𝑘
3- Calculate the moment of force F about point B:
⃗⃗⃗𝑏 = 𝑟⃗𝑏𝑐 × 𝐹⃗ = (−30𝑗 + 40𝑘) × (−2000𝑗 + 450𝑘) = 66,500𝑖
𝑀
4- Find the unit vector in the AB direction:
𝑢
⃗⃗𝑎𝑏 =
⃗⃗⃗⃗⃗⃗
𝐴𝐵
−40𝑖 + 30𝑗
=
= −0.8𝑖 + 0.6𝑗
|𝐴𝐵|
50
5- Find the component of the moment in the AB direction:
|𝑀𝑎𝑏 | = ⃗⃗⃗⃗⃗⃗
𝑀𝑎 ∙ 𝑢
⃗⃗𝑎𝑏 = (66,500𝑖) ∙ (−0.8𝑖 + 0.6𝑗) = −53,200 N-m
6- Express the component of the moment in the AB direction as a Cartesian vector:
⃗⃗⃗𝑎𝑏 = |𝑀𝑎𝑏 | 𝑢
𝑀
⃗⃗𝑎𝑏 = (−53,200)(−0.8𝑖 + 0.6𝑗) = 42,560𝑖 − 31,920𝑗
OR … point A could also be used as the reference for the moment. See next page.
1- Express force F as a Cartesian vector:
−40𝑗 + 9𝑘
𝐹⃗ = |𝐹| 𝑢
⃗⃗𝐹 = 2050 (
) = −2000𝑗 + 450𝑘
41
2- Find vector r:
𝑟⃗𝑎𝑐 = −40𝑖 + 40𝑘
3- Calculate the moment of force F about point A:
⃗⃗⃗𝑎 = 𝑟⃗𝑎𝑐 × 𝐹⃗ = (−40𝑖 + 40𝑘) × (−2000𝑗 + 450𝑘) = 80,000𝑖 + 18,000𝑗 + 80,000𝑘
𝑀
4- Find the unit vector in the AB direction:
𝑢
⃗⃗𝑎𝑏 =
⃗⃗⃗⃗⃗⃗
𝐴𝐵
−40𝑖 + 30𝑗
=
= −0.8𝑖 + 0.6𝑗
|𝐴𝐵|
50
5- Find the component of the moment in the AB direction:
⃗⃗⃗𝑎 ∙ 𝑢
|𝑀𝑎𝑏 | = 𝑀
⃗⃗𝑎𝑏 = (80,000𝑖 + 18000𝑗 + 80000𝑘) ∙ (−0.8𝑖 + 0.6𝑗) = −53,200 N-m
6- Express the component of the moment in the AB direction as a Cartesian vector:
⃗⃗⃗𝑎𝑏 = |𝑀𝑎𝑏 | 𝑢
𝑀
⃗⃗𝑎𝑏 = (−53,200)(−0.8𝑖 + 0.6𝑗) = 42,560𝑖 − 31,920𝑗
This is the same answer as when using reference point B on line AB