Question

two forces whose magnitude are in ratio3:5 gives resultant 35 N ,if theta =60° .Calculate magnitude of each force
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Answer 1

Diagram :

$\setlength{\unitlength}{1.2mm}\begin{picture}(5,5)\thicklines\put(0,61){ \sf{\overrightarrow{B}} }\put(60.8,0){ \sf{\overrightarrow{A}} }\put(50,55){ \sf{\overrightarrow{R}} }\put(0,5){\vector(0,0){56.5}}\put( - 0.1,5){\vector(1, 0){60.5}}\put(10,8){ \sf{60^{ \circ}} }\qbezier(0,5)(0,5)(50,55)\qbezier(0,15)(9.99,15)(9,5)\end{picture}$

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Given :

$\bullet \: \: \sf |\overrightarrow{R}| = 35 \: N$

$\bullet \: \: \sf \dfrac{|\overrightarrow{A}|}{|\overrightarrow{B}|} = \dfrac{3}{5}$

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Answer :

Let $\sf \overrightarrow{A}$ and $\sf \overrightarrow{B}$ are the two vectors inclined at an angle 60° and $\sf \overrightarrow{R}$ is resultant.

Let $\sf \overrightarrow{A}\:$ be 3x and $\sf \overrightarrow{B}$ be 5x.

☯ $\underline{\boldsymbol{According\: to \:the\: Question\:now :}}$

$:\implies \sf \overrightarrow{ |R| } = \sqrt{ \overrightarrow{{ |A| }} ^{2} + \overrightarrow{{ |B| } }^{2} + 2 \overrightarrow{ |A| } \: \overrightarrow{ |B| } \cos (\theta) }$

$:\implies \sf 35= \sqrt{(3x)^{2} +(5x)^{2} + 2(2x)(5x) \cos(60) }$

$:\implies \sf 35= \sqrt{9x^{2} +25x^{2} + {30x}^{2} \left( \frac{1}{2} \right) }$

$:\implies \sf 35= \sqrt{9x^{2} +25x^{2} + {15x}^{2} }$

$:\implies \sf 35= \sqrt{49x^{2} }$

$:\implies \sf 35= 7x$

$:\implies \underline{ \boxed{ \textsf{\textbf {x = 5}}}}$

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☯ $\underline{\boldsymbol{Magnitude\: of \:each\: Force\:is :}}$

$\bullet\:\:\textsf{ \overrightarrow{A} = 3(5) =\textbf{ 15 N}}$

$\bullet\:\:\textsf{ \overrightarrow{B} = 5(5) =\textbf{ 25 N}}$