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:
\(\ds \mathbf F_1\) | \(=\) | \(\ds \polar {100 \, \mathrm {kg}, 150 \degrees}\) | ||||||||||||
\(\ds \mathbf F_2\) | \(=\) | \(\ds \polar {75 \, \mathrm {kg}, 60 \degrees}\) | ||||||||||||
\(\ds \mathbf F_3\) | \(=\) | \(\ds \polar {50 \, \mathrm {kg}, -45 \degrees}\) |
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
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
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Polar Form of Complex Numbers: $86 \ \text {(a)}$