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The greatest and the least resultants of two forces acting at a point are F and G respectively. If x be the angle between the lines of action of the forces, then show that their resultant is given by
√F²cos²(x÷2)+G²sin²(x÷2)

Shams Ishtiaque Rahman , 7 Years ago

Grade 12th pass

1 Answers

Eshan

Last Activity: 6 Years ago

Let the vectors be $\vec{A}$ and $\vec{B}$ .

Then , $F=A+B$ and $G=A-B$

$\implies A=\dfrac{F+G}{2}$ and $B=\dfrac{F-G}{2}$

The resultant of the two vector is

$\sqrt{A^2+B^2+2ABcos x}$ $\sqrt{A^2+B^2+2ABcos x}=\sqrt{(\dfrac{F+G}{2})^2+(\dfrac{F-G}{2})^2+2(\dfrac{F+G}{2})(\dfrac{F-G}{2})cos x}$
$=\sqrt{\dfrac{F^2}{2}(1+cos x)+\dfrac{G^2}{2}(1-cos x)}$

Also we know $1+cos x=2cos^2(\dfrac{x}{2}); 1-cos x=2sin^2(\dfrac{x}{2})$

$\implies$ Resultant $=\sqrt{F^2cos^2(\dfrac{x}{2})+G^2sin^2(\dfrac{x}{2})}$

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