Question

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

• A. $F\sqrt{2\left(1-sin\,\theta \right)}$
• B. $F\sqrt{2\left(1+sin\,\theta \right)}$
• C. $2\,F\,sin \frac{\theta}{2}$
• D. $2\,F\,cos \left(\frac{\theta }{2}\right)$

Answer 1

Answer: (D)

Here, $F_1 = F_2 = F$
The magnitude of the resultant force is
$F_{R}=\sqrt{F^{2}_{1}+F^{2}_{2}+2F_{1}F_{2}\,cos\,\theta}=\sqrt{F^{2}+F^{2}+2\left(F\right)\left(F\right)cos\,\theta}$
$=\sqrt{2F^{2}+2F^{2}\,cos\,\theta }=F\sqrt{2\left(1+cos\,\theta \right)}$
$=F\sqrt{2\left(2\,cos^{2} \frac{\theta}{2}\right)}=2F\,cos\left(\frac{\theta}{2}\right)$