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  4. Two equal forces are acting at a point with an angle of 60° between them. If the resultant force is equal to 40√3N, the magnitude of each force is

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Q. Two equal forces are acting at a point with an angle of $ 60{}^\circ $ between them. If the resultant force is equal to $ 40\sqrt{3}N, $ the magnitude of each force is

CMC MedicalCMC Medical 2008

Solution:

Let equal forces $ {{F}_{1}}={{F}_{2}}=F\,N. $ Angle between the forces $ (\theta )=60{}^\circ $ Resultant force $ (R)=40\sqrt{3}N. $ Resultant force $ R=\sqrt{F_{1}^{2}+F_{2}^{2}+2{{F}_{1}}\,{{F}_{2}}\cos \theta } $ $ \therefore $ $ 40\sqrt{3}=\sqrt{{{F}^{2}}+{{F}^{2}}+2F.\,F\cos 60{}^\circ } $ or $ 40\sqrt{3}=\sqrt{2{{F}^{2}}+2{{F}^{2}}\times \frac{1}{2}} $ or $ 40\sqrt{3}=\sqrt{2{{F}^{2}}+{{F}^{2}}} $ or $ 40\sqrt{3}=\sqrt{3{{F}^{2}}} $ or $ 40\sqrt{3}=F\sqrt{3} $ or $ F=40\,N $