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Engineering Statics: Open and Interactive

Section 4.2 Magnitude of a Moment

As you probably know, the turning effect produced by a wrench depends on where and how much force you apply to the wrench, and the optimum direction to apply the force is at right angles to the wrench’s handle. If the nut won’t budge, you need to apply a larger force or get a longer wrench.
This strength of this turning effect is what we mean by the magnitude of a moment (or of a torque).

Subsection 4.2.1 Definition of a Moment

The magnitude of a moment is found by multiplying the magnitude of force \(\vec{F}\) times the moment arm, where the moment arm is defined as the perpendicular distance, \(d_{\perp}\text{,}\) from the center of rotation to the line of action of the force, measured perpendicularly as illustrated in the interactive.
\begin{equation} M =F d_{\perp}\text{.}\tag{4.2.1} \end{equation}


This interactive shows \(d_\perp\text{,}\) the perpendicular distance between the center of rotation and the line of action of the force. The moment is the force times the perpendicular distance.
Figure 4.2.1. Definition of the moment, \(M = F d_\perp\text{.}\)
Notice that the magnitude of a moment depends only on the force and the moment arm, so the same force produces different moments about different points in space. The closer the center of rotation is to the force’s line of action, the smaller the moment. Points on the force’s line of action experience no moment because there the moment arm is zero. Furthermore, vector magnitudes are always positive, so clockwise and counter-clockwise moments with the same strength have the same magnitude.