Statement I: Two forces $(\vec P + \vec Q)$ and $(\vec P - \vec Q)$ where $\vec P \perp \vec Q$, when act at an angle $\theta_1$ to each other, the magnitude of their resultant is $\sqrt{3(P^2 + Q^2)}$; when they act at an angle $\theta_2$, the magnitude of their resultant becomes $\sqrt{2(P^2 + Q^2)}$. This is possible only when $\theta_1 < \theta_2$.
Statement II: In the situation given above, $\theta_1 = 60^\circ$ and $\theta_2 = 90^\circ$.
In the light of the above statements, choose the most appropriate answer from the options given below:
Answer: (B) Both Statement I and Statement II are true.
Since $\vec P \perp \vec Q$, both forces have magnitude $F = \sqrt{P^2 + Q^2}$.
Resultant of two equal forces at angle $\theta$: $R^2 = 2F^2(1 + \cos\theta)$.
$R^2 = 3F^2 \Rightarrow \cos\theta_1 = \dfrac12 \Rightarrow \theta_1 = 60^\circ$; $R^2 = 2F^2 \Rightarrow \cos\theta_2 = 0 \Rightarrow \theta_2 = 90^\circ$.
So $\theta_1 < \theta_2$ and both statements are true.
Solution by Sreeraj P, M.Sc Physics