Four identical particles of equal masses 1kg made to move along the...

${F}_{1} = \dfrac{Gmm}{{\left(2R\right)}^{2}} = \dfrac{G{m}^{2}}{4{R}^{2}}$

${F}_{2} = \dfrac{Gmm}{{\left(\sqrt{2}R\right)}^{2}} = \dfrac{G{m}^{2}}{2{R}^{2}}$

${F}_{3} = \dfrac{Gmm}{{\left(\sqrt{2}R\right)}^{2}} = \dfrac{G{m}^{2}}{2{R}^{2}}$

$⇒ {F}_{net} = {F}_{1} + {F}_{2}\cos 45 ° + {F}_{3}\cos 45 °$

$= \dfrac{G{m}^{2}}{4{R}^{2}} + \dfrac{G{m}^{2}}{2{R}^{2}}\dfrac{1}{\sqrt{2}} + \dfrac{G{m}^{2}}{2{R}^{2}}\dfrac{1}{\sqrt{2}}$

$= \dfrac{G{m}^{2}}{{R}^{2}}\left(\dfrac{1}{4} + \dfrac{1}{\sqrt{2}}\right) = \dfrac{G{m}^{2}}{4{R}^{2}}\left(1 + 2\sqrt{2}\right)$

${F}_{net} = \dfrac{G{m}^{2}}{4{R}^{2}}\left(1 + 2\sqrt{2}\right) = \dfrac{m{v}^{2}}{R}$

$⇒ v = \dfrac{\sqrt{G\left(1 + 2\sqrt{2}\right)}}{2}$