A force of F=\left\{6i-2j+1k\right\} kN produces a moment of M_o=\left\{4i+5j-14k\right\}\,\text{kN}\cdot\text{m} about the origin of coordinates, point O. If the force acts at a point having an x coordinate of x = 1 m, determine the y and z coordinates.
Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.
Solution:
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The moment is found by taking the cross product between a position vector from the origin and the force. Thus, we can write:
M_o=r\times F
Substitute the values we know:
\left\{4i+5j-14k\right\}=\begin{bmatrix}\bold i&\bold j&\bold k\\1&y&z\\6&-2&1\end{bmatrix}
\left\{4i+5j-14k\right\}=(y+2z)i-(1-6z)j+(-2-6y)k
\left\{4i+5j-14k\right\}=(y+2z)i-(1-6z)j+(-2-6y)k
Separating the components gives us:
4=y+2z
5=-1+6z
-14=-2-6y
5=-1+6z
-14=-2-6y
Solving the equations gives us:
y=2 m
z=1 m
z=1 m
Final Answers:
y=2 m
z=1 m
z=1 m