Question

The resultant of two forces $3P$ and $2P$ is $R$. If the first force is doubled, then the resultant is also doubled. The angle between the two forces is

• A. $120^{\circ }$
• B. $60^{\circ }$
• C. $180^{\circ }$
• D. $90^{\circ }$

Answer 1

Answer: (A)

$R^2=(3P{)}^2+(2P{)}^2+2(3P)(2P)\cos \theta$
$\Rightarrow R^2=13P^2+12P^2\cos \theta \dots (1)$
When the first force is doubled, the resultant is doubled,
so, $(2R{)}^2=(6P{)}^2+(2P{)}^2+2(6P)(2P)\cos \theta$
$\Rightarrow 4R^2=36P^2+4P^2+24P^2\cos \theta$
$\Rightarrow R^2=10P^2+6P^2\cos \theta \dots (2)$
Equating (1) & (2), we get
$13P^2+12P^2\cos \theta =10P^2+6P^2\cos \theta$
$3P^2=-6P^2\cos \theta$
so, $\cos \theta =-\frac{1}{2}$
$\Rightarrow \theta =120^{\circ }$