Question

Two force $P$ and $Q$ act at such an angle that $R=P$. Show that if $P$ is doubled, the new resultant is at right angles to $Q$.

Answer 1

Let the angle between the forces $P$ and $Q$ is $\alpha$.

The resultant force is given as,

$P^2=P^2+Q^2+2PQ\cos \alpha$

$\cos \alpha =-\frac{Q}{2P}$

When the $P$ is doubled, let the new resultant make an angle $\theta$ with $Q$.

It can be written as,

$\tan \theta =\frac{2P\sin \alpha }{Q+2P\cos \alpha }$

$=\frac{2P\sin \alpha }{Q+2P\left(-\frac{Q}{2P}\right)}$

$\tan \theta =\infty$

$\theta =\frac{\pi }{2}$

Thus, the new resultant is at right angles to $Q$.