Question

three masses, each equal to M are placed at the three corners of a square of side a. calculate the force of attraction on unit mass placed at the fourth corner.​

Answer 1

$\huge{\mathbf{\red{\underline{\underline{Solution:-}}}}}$

$\mathtt{\underline{Newton's\: universal\:law\:of\: Gravitation:}}$

$F= \frac{Gm_1m_2}{ {r}^{2} }$

$\mathtt{\underline{Force\: on\:mass\:at\:D\:due\:to\:mass\:at\:A:}}$

$\mathtt{F_{1}=\frac{Gm^{2}}{L^{2}}}$

$\mathtt{\underline{Force\: on\:mass\:at\:D\:due\:to\:mass\:at\:C:}}$

$\mathtt{F_{2}=\frac{Gm^{2}}{L^{2}}}$

$\mathtt{\underline{Force\: on\:mass\:at\:D\:due\:to\:mass\:at\:B:}}$

$\mathtt{F_{3}=\frac{Gm^{2}}{2L^{2}}}$

$\mathtt{\underline{Net\: force\:at\:D\:due\:to\:A\:and\:C:}}$

$\mathtt{F_{4}= \sqrt{F_{1}^{2}+F_{2}^{2}}}$

$\mathtt{=\sqrt{2}\frac{Gm^{2}}{L^{2}}}$

$\mathtt{\underline{Net\: force\:on\:D:}}$

$\mathtt{F_{net}=F_{4}+F_{3}}$

$\mathtt{=\sqrt{2}\frac{Gm^{2}}{L^{2}}}+\frac{Gm^{2}}{2L^{2}}$

$\mathtt{\boxed{=\frac{Gm^{2}}{L^{2}}}}$

Answer 2

$\huge\bf{\pink{\mid{\underline{\overline{your\: answer}}}\mid}}$

$\large{\underline{\underline{\mathbf{Given:-}}}}$

Three masses each equal to M.

M = 1

Three corners of a square of side A.

$f_{1} \: \: and \: \: \: f_{3}$

$\large{\underline{\underline{\mathbf{To \:find-}}}}$

✡ Force of attraction of unit mass at the fourth corner.

$\large{\underline{\underline{\mathbf{Solution-}}}}$

Force on M = 1

✏ at corners 1 and 3,

$f_{1} = f_{3} = \frac{gm}{ {a}^{2} }$

✏ resultant of these F1 and F3 vectors are,

$f_{r} = \sqrt{2} \frac{gm}{ {a}^{2} }$

and it's direction is along the daigonal.

★$\bold{i.e.\:toward\: corner \:2}$

Then,

Force on m due to M at 2 is

$f_{2} = \frac{gm}{ { \sqrt{2a} }^{2} } = \frac{gm}{ {2a}^{2} }$

★$\bold{Fr\:and\: F2\: act\: in\: the \: same \: direction}$

$\rm{\underline{Resultant\: of\: these\: two\: net\: force:}}$

$= > f_{net} = \frac{ \sqrt{2gm {a}^{2} } }{ {a}^{2} } + \frac{gm}{ {a}^{2} } ( \sqrt{2} + \frac{1}{2} )$

★$\bold{it\: is\: directed\: along\: the\: daigonal}$

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