Question

The resultant of forces $P$ and $Q$ is $R$. If $Q$ is doubled, then $R$ is doubled. If the direction of $Q$ is reversed,then $R$ is again doubled. Then $P^2:Q^2:R^2=$

• A. $3:1:1$
• B. $2:3:2$
• C. $1:2:3$
• D. $2:3:1$

Answer 1

Answer: (B)

$2:3:2$

Given: $|\overrightarrow{P}-\overrightarrow{Q}|=2\left|\overrightarrow{R}\right|$
$\Rightarrow P^2+Q^2-2.\overrightarrow{P}\overrightarrow{Q}=4R^2$ ..........$\left(3\right)$
Eqns $\left(1\right)+\left(3\right)=2P^2+2Q^2=5R^2$
$\Rightarrow 2P^2+2Q^2-5R^2=0$ ............. $\left(4\right)$
Eqns $\left(2\right)+2\times \left(3\right)$
$\Rightarrow 3P^2+6Q^2-12R^2=0$
or $P^2+2Q^2-4R^2=0$ ............$\left(5\right)$
By eliminating $\left(4\right)$ and $\left(5\right)$ we get
$\frac{P^2}{2}=\frac{Q^2}{3}=\frac{R^2}{2}$
$\therefore$ the ratio is $2:3:2$