4.2: Magnitude of a Moment

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Key Questions

• Why is there no moment about any point on the line of action of a force?
• If you increase the distance between a force and a point of interest, does the moment of the force go up or down?
• What practical applications can you think of that could use moments to describe?

Moments in two dimensions can only cause a clockwise or counter-clockwise rotation in the plane of the page. This means that we can completely describe a two-dimensional moment with a signed scalar value. This signed value is called the scalar moment. The numerical value indicates its magnitude and the sign indicates its direction. By convention, a positive sign indicates a counter-clockwise moment and a negative sign means clockwise. This sign convention is chosen to be consistent with the right hand rule.

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

The magnitude of a scalar moment can be 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{.}\label{moment-definition}\tag{4.2.1} \end{equation}

To this value you must attach a positive sign to indicate counter-clockwise rotation or a negative sign to indicate clockwise rotation.

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 since the magnitude of a moment depends only on the force and the moment arm, different points on the wrench experience different moments. Points on the force’s line of action experience no moment because there the moment arm is zero.