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Physics Question on Newtons Laws of Motion

Two forces of equal magnitude FF act at a point. If the angle between them is ??, then the magnitude of the resultant force is

A

F2(1−sin θ)F\sqrt{2\left(1-sin\,\theta \right)}

B

F2(1+sin θ)F\sqrt{2\left(1+sin\,\theta \right)}

C

2 F sinθ22\,F\,sin \frac{\theta}{2}

D

2 F cos(θ2)2\,F\,cos \left(\frac{\theta }{2}\right)

Answer

2 F cos(θ2)2\,F\,cos \left(\frac{\theta }{2}\right)

Explanation

Solution

Here, F1=F2=FF_1 = F_2 = F The magnitude of the resultant force is FR=F12+F22+2F1F2 cos θ=F2+F2+2(F)(F)cos θF_{R}=\sqrt{F^{2}_{1}+F^{2}_{2}+2F_{1}F_{2}\,cos\,\theta}=\sqrt{F^{2}+F^{2}+2\left(F\right)\left(F\right)cos\,\theta} =2F2+2F2 cos θ=F2(1+cos θ)=\sqrt{2F^{2}+2F^{2}\,cos\,\theta }=F\sqrt{2\left(1+cos\,\theta \right)} =F2(2 cos2θ2)=2F cos(θ2)=F\sqrt{2\left(2\,cos^{2} \frac{\theta}{2}\right)}=2F\,cos\left(\frac{\theta}{2}\right)