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Question Number 95726 by Rio Michael last updated on 27/May/20
th forces F_1  ,F_2 ,F_3  act at points ith position vectors r_1 ,r_2 ,r_3   where  F_1  = (4i + j + 2k)N       r_1  = (6i + 4j + k) m  F_2  = (i−2j + k)N           r_2  = (i + 5j −2k) m   F_3  = (−5i + j−3k)N    r_3  = (i + j + k) m  (a) show that this system reduces to a couple and find its magnitude.  (b) if F_3  is removed and replaced with F_4   such that the system is   now in equilibrium find the force vector F_4 .  (c) the equation of the line of action of F_4   (d) the moment of F_4  about the origin.
$$\mathrm{th}\:\mathrm{forces}\:{F}_{\mathrm{1}} \:,{F}_{\mathrm{2}} ,{F}_{\mathrm{3}} \:\mathrm{act}\:\mathrm{at}\:\mathrm{points}\:\mathrm{ith}\:\mathrm{position}\:\mathrm{vectors}\:{r}_{\mathrm{1}} ,{r}_{\mathrm{2}} ,{r}_{\mathrm{3}} \:\:\mathrm{where} \\ $$$${F}_{\mathrm{1}} \:=\:\left(\mathrm{4}{i}\:+\:{j}\:+\:\mathrm{2}{k}\right){N}\:\:\:\:\:\:\:{r}_{\mathrm{1}} \:=\:\left(\mathrm{6}{i}\:+\:\mathrm{4}{j}\:+\:{k}\right)\:\mathrm{m} \\ $$$${F}_{\mathrm{2}} \:=\:\left({i}−\mathrm{2}{j}\:+\:{k}\right){N}\:\:\:\:\:\:\:\:\:\:\:{r}_{\mathrm{2}} \:=\:\left({i}\:+\:\mathrm{5}{j}\:−\mathrm{2}{k}\right)\:\mathrm{m} \\ $$$$\:{F}_{\mathrm{3}} \:=\:\left(−\mathrm{5}{i}\:+\:{j}−\mathrm{3}{k}\right){N}\:\:\:\:{r}_{\mathrm{3}} \:=\:\left({i}\:+\:{j}\:+\:{k}\right)\:\mathrm{m} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{show}\:\mathrm{that}\:\mathrm{this}\:\mathrm{system}\:\mathrm{reduces}\:\mathrm{to}\:\mathrm{a}\:\mathrm{couple}\:\mathrm{and}\:\mathrm{find}\:\mathrm{its}\:\mathrm{magnitude}. \\ $$$$\left(\mathrm{b}\right)\:\mathrm{if}\:\mathrm{F}_{\mathrm{3}} \:\mathrm{is}\:\mathrm{removed}\:\mathrm{and}\:\mathrm{replaced}\:\mathrm{with}\:{F}_{\mathrm{4}} \:\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{system}\:\mathrm{is}\: \\ $$$$\mathrm{now}\:\mathrm{in}\:\mathrm{equilibrium}\:\mathrm{find}\:\mathrm{the}\:\mathrm{force}\:\mathrm{vector}\:{F}_{\mathrm{4}} . \\ $$$$\left(\mathrm{c}\right)\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{line}\:\mathrm{of}\:\mathrm{action}\:\mathrm{of}\:{F}_{\mathrm{4}} \\ $$$$\left(\mathrm{d}\right)\:\mathrm{the}\:\mathrm{moment}\:\mathrm{of}\:\mathrm{F}_{\mathrm{4}} \:\mathrm{about}\:\mathrm{the}\:\mathrm{origin}. \\ $$
Commented by Rio Michael last updated on 29/May/20
please  anyone help me with this Q95726
$$\mathrm{please}\:\:\mathrm{anyone}\:\mathrm{help}\:\mathrm{me}\:\mathrm{with}\:\mathrm{this}\:\mathrm{Q95726} \\ $$

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