Magnitude and Direction of Equilibrant

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Theorem

Let $\mathbf F_1, \mathbf F_2, \ldots, \mathbf F_n$ be a set of $n$ forces acting on a particle $B$ at a point $P$ in space.


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

$\mathbf E = -\ds \sum_{k \mathop = 1}^n \mathbf F_k$


That is, the magnitude and direction of $\mathbf E$ is such as to balance out the effect of $\mathbf F_1, \mathbf F_2, \ldots, \mathbf F_n$.


Proof

From Newton's First Law of Motion, the total force on $B$ must equal zero in order for $B$ to remain stationary.


That is, $\mathbf E$ must be such that:

$\mathbf E + \ds \sum_{k \mathop = 1}^n \mathbf F_k = \bszero$

That is:

$\mathbf E = -\ds \sum_{k \mathop = 1}^n \mathbf F_k$

$\blacksquare$