Question

If the resultant of two forces $P$ and $Q$ be equal in magnitude to one of the components $P$ and perpendicular to it in direction, then the value of $Q$ is

• A. $P$
• B. $\frac{1}{\sqrt{2}}P$
• C. $\sqrt{2}P$
• D. $\sqrt{3}P$

Answer 1

Answer: (C)

$\sqrt{2}P$

Let, $\overrightarrow{R}=\overrightarrow{P}+\overrightarrow{Q}R=P\overrightarrow{R}\perp \overrightarrow{P}\overrightarrow{R}\cdot \overrightarrow{P}=0\Rightarrow (\overrightarrow{P}+\overrightarrow{Q})\cdot \overrightarrow{P}=0\Rightarrow P^2+\overrightarrow{Q}\cdot \overrightarrow{P}=0\left(1\right)R^2=Q^2+P^2+2\overrightarrow{Q}\cdot \overrightarrow{P}=P^2\Rightarrow Q^2+2\overrightarrow{Q}\cdot \overrightarrow{P}=0\left(2\right)$

From $\left(1\right)$ and $ \left( 2 \right) :

$Q^2=2P^2\Rightarrow Q=\sqrt{2}P$