Equilibrant/Examples/100kg at 150, 75kg at 60, 50kg at -45/Proof 2

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Example of Equilibrant

Three forces $\mathbf F_1, \mathbf F_2, \mathbf F_3$ act on a particle $B$ at a point $P$ embedded in the complex plane:

\(\displaystyle \mathbf F_1\) \(=\) \(\displaystyle \polar {100 \, \mathrm {kg}, 150 \degrees}\)
\(\displaystyle \mathbf F_2\) \(=\) \(\displaystyle \polar {75 \, \mathrm {kg}, 60 \degrees}\)
\(\displaystyle \mathbf F_3\) \(=\) \(\displaystyle \polar {50 \, \mathrm {kg}, -45 \degrees}\)


Equilibrant-100kg at 150, 75kg at 60, 50kg at -45.png


The equilibrant $\mathbf E$ of $\mathbf F_1, \mathbf F_2, \mathbf F_3$ is:

$\mathbf E = \polar {80.8 \, \mathrm {kg}, -80.2 \degrees}$


Proof

Equilibrant-100kg at 150, 75kg at 60, 50kg at -45-Solution.png

Using the Parallelogram Law to add $\mathbf F_1$ and $\mathbf F_2$ we arrive at $\mathbf F_1 + \mathbf F_2$.

Again using the Parallelogram Law to add $\mathbf F_1 + \mathbf F_2$ and $\mathbf F_3$ we arrive at $\mathbf F_1 + \mathbf F_2 + \mathbf F_3$.

The equilibrant $\mathbf E$ can then be found by taking the negative of $\mathbf F_1 + \mathbf F_2 + \mathbf F_3$.

$\blacksquare$


Sources