Question

# Replace the 880-lb force acting at point A by a force-couple system at (a) point O and (b) point B. 11510200791.jpg

Complex numbers
Replace the 880-lb force acting at point A by a force-couple system at
(a) point O
and
(b) point B.

2020-11-09
$$\displaystyle{o}{v}{e}{r}\rightarrow{\left\lbrace{F}\right\rbrace}={800}\ {l}{b}\ {\left[-{\frac{{{2}}}{{\sqrt{{{3}}}}}}{w}{i}{d}{e}\hat{{{i}}}+{\frac{{{1}}}{{\sqrt{{{3}}}}}}{w}{i}{d}{e}\hat{{{i}}}\right]}$$
a) Point O
$$\displaystyle\Rightarrow{o}{v}{e}{r}\rightarrow{\left\lbrace{O}{A}\right\rbrace}={3}{j}$$
$$\displaystyle\therefore{o}{v}{e}{r}\rightarrow{\left\lbrace{C}\right\rbrace}={o}{v}{e}{r}\rightarrow{\left\lbrace{O}{A}\right\rbrace}\times{o}{v}{e}{r}\rightarrow{\left\lbrace{F}\right\rbrace}$$
$$\displaystyle={3}{j}\times{800}\ {l}{b}\ {\left[-{\frac{{{2}}}{{\sqrt{{{3}}}}}}{w}{i}{d}{e}\hat{{{i}}}+{\frac{{{1}}}{{\sqrt{{{3}}}}}}{w}{i}{d}{e}\hat{{{i}}}\right]}$$
$$\displaystyle={2771}\cdot{28}{w}{i}{d}{e}\hat{{{k}}}\ {l}{b}-{f}{t}$$
b) Point B $$\displaystyle\Rightarrow{o}{v}{e}{r}\rightarrow{\left\lbrace{B}{A}\right\rbrace}={4}{i}+{3}{w}{i}{d}{e}\hat{{{j}}}$$
$$\displaystyle\therefore{o}{v}{e}{r}\rightarrow{\left\lbrace{C}\right\rbrace}={o}{v}{e}{r}\rightarrow{\left\lbrace{B}{A}\right\rbrace}\times{o}{v}{e}{r}\rightarrow{\left\lbrace{F}\right\rbrace}$$
$$\displaystyle={\left({4}{w}{i}{d}{e}\hat{{{i}}}+{3}{w}{i}{d}{e}\hat{{{j}}}\right)}\times{800}\ {l}{b}{\left[-{\frac{{{2}}}{{\sqrt{{{3}}}}}}{w}{i}{d}{e}\hat{{{i}}}+{\frac{{{1}}}{{\sqrt{{{3}}}}}}{w}{i}{d}{e}\hat{{{i}}}\right]}$$
$$\displaystyle={4618}\cdot{8}{w}{i}{d}{e}\hat{{{k}}}\ {l}{b}-{f}{t}$$