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Question Number 122189 by physicstutes last updated on 14/Nov/20
Given that the forces F_1 , F_2  and F_3  have position vectors   r_1 , r_2  and r_3 . Where,   F_1  = (2i + 3i + k) N      r_1  = (i + 2k) m    F_2  = (3i + 2i + 5k) N    r_2  = (2i − 4j + k) m   F_3  = (−5i−5j −6k) N    r_3  = (4i + 3j + 2k) m  (a) Find the moment of F_1  about (2i + 3j + 4k) m.  (b) Show that this system of forces reduces to a couple, G,   and find the magnitude of this couple.  (c) Check if F_1  and F_2  are concurrent.
$$\mathrm{Given}\:\mathrm{that}\:\mathrm{the}\:\mathrm{forces}\:\boldsymbol{\mathrm{F}}_{\mathrm{1}} ,\:\boldsymbol{\mathrm{F}}_{\mathrm{2}} \:\mathrm{and}\:\boldsymbol{\mathrm{F}}_{\mathrm{3}} \:\mathrm{have}\:\mathrm{position}\:\mathrm{vectors} \\ $$$$\:\boldsymbol{\mathrm{r}}_{\mathrm{1}} ,\:\boldsymbol{\mathrm{r}}_{\mathrm{2}} \:\mathrm{and}\:\boldsymbol{\mathrm{r}}_{\mathrm{3}} .\:\mathrm{Where}, \\ $$$$\:\boldsymbol{\mathrm{F}}_{\mathrm{1}} \:=\:\left(\mathrm{2}\boldsymbol{\mathrm{i}}\:+\:\mathrm{3}\boldsymbol{\mathrm{i}}\:+\:\boldsymbol{\mathrm{k}}\right)\:\mathrm{N}\:\:\:\:\:\:\boldsymbol{\mathrm{r}}_{\mathrm{1}} \:=\:\left(\boldsymbol{\mathrm{i}}\:+\:\mathrm{2}\boldsymbol{\mathrm{k}}\right)\:\mathrm{m} \\ $$$$\:\:\boldsymbol{\mathrm{F}}_{\mathrm{2}} \:=\:\left(\mathrm{3}\boldsymbol{\mathrm{i}}\:+\:\mathrm{2}\boldsymbol{\mathrm{i}}\:+\:\mathrm{5}\boldsymbol{\mathrm{k}}\right)\:\mathrm{N}\:\:\:\:\boldsymbol{\mathrm{r}}_{\mathrm{2}} \:=\:\left(\mathrm{2}\boldsymbol{\mathrm{i}}\:−\:\mathrm{4}\boldsymbol{\mathrm{j}}\:+\:\boldsymbol{\mathrm{k}}\right)\:\mathrm{m} \\ $$$$\:\boldsymbol{\mathrm{F}}_{\mathrm{3}} \:=\:\left(−\mathrm{5}\boldsymbol{\mathrm{i}}−\mathrm{5}\boldsymbol{\mathrm{j}}\:−\mathrm{6}\boldsymbol{\mathrm{k}}\right)\:\mathrm{N}\:\:\:\:\boldsymbol{\mathrm{r}}_{\mathrm{3}} \:=\:\left(\mathrm{4}\boldsymbol{\mathrm{i}}\:+\:\mathrm{3}\boldsymbol{\mathrm{j}}\:+\:\mathrm{2}\boldsymbol{\mathrm{k}}\right)\:\mathrm{m} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{moment}\:\mathrm{of}\:\boldsymbol{\mathrm{F}}_{\mathrm{1}} \:\mathrm{about}\:\left(\mathrm{2}\boldsymbol{\mathrm{i}}\:+\:\mathrm{3}\boldsymbol{\mathrm{j}}\:+\:\mathrm{4}\boldsymbol{\mathrm{k}}\right)\:\mathrm{m}. \\ $$$$\left(\mathrm{b}\right)\:\mathrm{Show}\:\mathrm{that}\:\mathrm{this}\:\mathrm{system}\:\mathrm{of}\:\mathrm{forces}\:\mathrm{reduces}\:\mathrm{to}\:\mathrm{a}\:\mathrm{couple},\:\boldsymbol{\mathrm{G}},\: \\ $$$$\mathrm{and}\:\mathrm{find}\:\mathrm{the}\:\mathrm{magnitude}\:\mathrm{of}\:\mathrm{this}\:\mathrm{couple}. \\ $$$$\left(\mathrm{c}\right)\:\mathrm{Check}\:\mathrm{if}\:\boldsymbol{\mathrm{F}}_{\mathrm{1}} \:\mathrm{and}\:\boldsymbol{\mathrm{F}}_{\mathrm{2}} \:\mathrm{are}\:\mathrm{concurrent}. \\ $$

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