Four coplanar forces of magnitudes $$$40 \mathrm{~N}, 20 \mathrm{~N}, 50 \mathrm{~N}$$$ and $$$F \mathrm{~N}$$$ act at a point in the directions shown in the diagram. The four forces are in equilibrium. Find $$$F$$$ and $$$\alpha$$$. ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_s20_qp_43 Year：2020 Question No：3

Answer:

Attempt to resolve, either direction with correct number of terms
$
F \cos \alpha=40 \sin 30+20 \sin 60-50 \sin 45(=1.965 \ldots)
$$
F \sin \alpha=50 \cos 45+20 \cos 60-40 \cos 30(=10.714 \ldots)
$
Method for either F or $\alpha$
$
\begin{array}{l}
F=\sqrt{\left((1.965 \ldots)^{2}+(10.714 \ldots)^{2}\right)}=10.9(10.893) \\
\alpha=\tan ^{-1}(10.714 \ldots / 1.965 \ldots)=79.6(79.606 \ldots)
\end{array}
$

Knowledge points:

4.1.2 understand the vector nature of force, and find and use components and resultants； Calculations are always required, not approximate solutions by scale drawing.

4.1.3 use the principle that, when a particle is in equilibrium, the vector sum of the forces acting is zero, or equivalently, that the sum of the components in any direction is zero （Solutions by resolving are usually expected, but equivalent methods (e.g. triangle of forces, Lami’s Theorem, where suitable) are also acceptable; these other methods are not required knowledge, and will not be referred to in questions.）

Answer:

Knowledge points:

Solution:

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