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. ${60}^{\circ }$
• B. ${120}^{\circ }$
• C. ${70}^{\circ }$
• D. ${180}^{\circ }$

Answer 1

The correct option is B ${120}^{\circ }$

In $1st$ case $R^2={\left(3P\right)}^2+{\left(2P\right)}^2+2\times 3\times 2\times P^2\cos \theta$

$={9P}^2+{4P}^2+12P^2\cos \theta$

$R^2={13P}^2+{12P}^{2}cos\theta ⟶\left(1\right)$

In $2nd$ case

${\left(2R\right)}^2={\left(6P\right)}^2+{\left(2P\right)}^2+2\times 6\times 2\times P^2\cos \theta$

$\Rightarrow {4R}^2={36P}^2+{4P}^2+24P^{2}cos\theta ⟶\left(2\right)$

Multiplying $\left(1\right)$ by $4$

${4R}^2={5LP}^2+4r{P}^2\cos \theta ⟶\left(3\right)$

$\left(3\right)$ $-$ $\left(2\right)$

$0={12P}^2+{24P}^2\cos \theta$

$\cos \theta =\frac{-1}{2}$

$\theta ={\cos }^2\left(\frac{-1}{2}\right)={120}^0$.