Question

The resultant of two forces $P$ and $Q$ is of magnitude $P$. If the force $P$ is doubled, $Q$ remaining the same, then angle between new resultant and the force $Q$ is

• A. ${60}^o$
• B. ${90}^o$
• C. ${30}^o$
• D. ${45}^o$

Answer 1

The correct option is B ${90}^o$

Let $\alpha$ be the angle between two forces $P$ and $Q$. It is given that the resultant of $P$ and $Q$ is of magnitude $P$.
Therefore, $P^2=P^2+Q^2+2PQ\cos \alpha$
$\Rightarrow Q(Q+2P\cos \alpha )=0$
$\Rightarrow Q+2P\cos \alpha =0$
$2P\cos \alpha =-Q$ ...(i)
Let $\theta$ be the angle between the force $Q$ and new resultant.
Then, $\tan \theta =\frac{2P\sin \alpha }{Q+2P\cos \alpha }$
$\Rightarrow \tan \theta =\frac{2P\sin \alpha }{Q-Q}=\infty$ [from equation (i)]
$\Rightarrow \theta =\frac{\pi }{2}$.