Question

Two forces, each of magnitude ${}^{\prime }{f}^{\prime }$ are making an angle $\varphi$ act on a body, then the resultant force is:

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

Answer 1

The correct option is D $2f\cos (\frac{\varphi }{2})$

We know that Resultant, $R=\sqrt{f^2+f^2+2f^{2}cos\varphi }$
which gives, $R=\sqrt{2f^2(1+cos\varphi )}$
Now, $cos2\theta =2co{s}^2\theta -1$

Hence, $cos\theta =2co{s}^2\left(\theta /2\right)-1$ or $cos\theta +1=2co{s}^2\theta /2$
Thus, $R=\sqrt{2f^2\left(2co{s}^2\right(\varphi /2\left)\right)}=2fcos\left(\varphi /2\right)$