The sum of two forces is 18 N and the resultant, whose direction is at right angles to the…

Question

The sum of two forces is 18 N and the resultant, whose direction is at right angles to the smaller force, is 12 N. The magnitudes of the two forces are:

Accepted Answer

Let the forces be $P$ (larger) and $Q$ (smaller), $P+Q=18$, with resultant $R=12$ perpendicular to $Q$.

Perpendicularity to $Q$ means the component of $P$ along $Q$ cancels $Q$: $P\cos\alpha+Q=0 \implies \cos\alpha=-\dfrac QP$, and $R=P\sin\alpha$.

Testing $P=13,Q=5$: $\cos\alpha=-\dfrac5{13}$, $\sin\alpha=\dfrac{12}{13}$, $R=13\times\dfrac{12}{13}=12$ , and $P+Q=18$ .

$$\text{Forces are } 13\text{ N and }5\text{ N}$$

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