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Q.
Three forces of magnitude $ 30 $ , $ 60 $ and $ P $ acting at a point are in equilibrium. If the angle between the first two is $ 60^{\circ} $ , the value of $ P $ is
Resultant of two forces $P$ and $Q$ making an angle $α$ with each other, is
$R=\sqrt{P^{2}+Q^{2}+2\,PQ\,cos\,\alpha}$
Since, forces of magnitude $30$, $60$ and $P$ are in equilibrium, then magnitude of $P$ is equal to the magnitude of resultant of forces of magnitude $30$ and $60$
$\therefore P=\sqrt{\left(30\right)^{2}+\left(60\right)^{2}+2\left(30\right)\left(60\right)\,cos\,60^{\circ}}$
$=\sqrt{900+3600+3600\times\frac{1}{2}}$
$=\sqrt{4500+1800}=\sqrt{6300}$
$=30\sqrt{7}$