Resolution of Forces
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Find the magnitude of the force, acting on a smooth inclined plane of angle \(30^\circ\), required to support a mass of 20kg resting on a plane.
$$\begin{aligned}
&\text{Resolving horizontally}\\
&N \cos 60^{\circ}=F \cos 30^{\circ} \ldots(1)\\
&\text{Resolving vertically}\\
&\begin{array}{l}
N \cos 30^{\circ}+F \cos 60^{\circ}=20 \ldots \text { (2) } \\
\text {in (1) }\ N=\frac{F \cos 30^{\circ}}{\cos 60^{\circ}} \\
\text { Sub in (2) } F \frac{\cos ^{2} 30^{\circ}}{\cos 60^{\circ}}+F \cos 60^{\circ}=20
\end{array}\\
&\begin{aligned}F \frac{\left(\cos ^{2} 30+\cos ^{2} 60\right)}{\cos 60^{\circ}}&=20\\
F\left(\cos ^{2} 30+\sin ^{2} 30\right)&=20 \times \cos 60^{\circ}\\
F&=20 \times \frac{1}{2}\\
F&=10 \mathrm{~kgwt}\end{aligned}
\end{aligned}$$
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