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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)

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)

Grade:12th pass

1 Answers

Eshan
askIITians Faculty 2095 Points
5 years ago
Let the vectors be\vec{A}and\vec{B}.

Then ,F=A+BandG=A-B

\implies A=\dfrac{F+G}{2}andB=\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 know1+cos x=2cos^2(\dfrac{x}{2}); 1-cos x=2sin^2(\dfrac{x}{2})

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

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