Question

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

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

Answer 1

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

Let the angle between the vectors $3P$ and $2P$ be $\theta$.
Then,
$R=\sqrt{\left(9P^2\right)+4P^2+2\times 3P\times 2P\times \cos \theta }$

Squaring both sides:
$R^2=13P^2+12P^2\cos \theta ....\left(i\right)$

When $3P\to 6P$, then $R\to 2R$, i.e resultant doubles.
$\Rightarrow 2R=\sqrt{\left(36P^2\right)+4P^2+2\times 6P\times 2P\times \cos \theta }$

Squaring both the sides:
$4R^2=40P^2+24P^2\cos \theta .....\left(ii\right)$

Putting value of $R^2$ from Eq. $\left(i\right)$ in Eq. $\left(ii\right)$:
$52P^2+48P^2\cos \theta =40P^2+24P^2\cos \theta$
$\Rightarrow 12P^2+24P^2\cos \theta =0$
$\Rightarrow \cos \theta =-\frac{1}{2}$
$\therefore \theta ={120}^{\circ }$