Section 4.2 Scalar Moments
Key Questions
At the end of this chapter you should be able to answer these questions.
- Why is there no moment about any point on the line of action of a force?
- If you increase the distance between your force and point of interest, does the moment of the force go up or down?
- What practical applications can you think of that could use this relationship?
As you likely already know or observed, the moment of a force depends on how far the force acts from the point about which you are taking the moment. In Figure DD below, the distance information is added to the wrench example outlined in Figure AA. The magnitude of the moment about A \(\vec{M_A}\) is calculated by multiplying the magnitude of force \(\vec{F}\) by the length of the perpendicular moment arm \(d_{\perp}\text{.}\) The scalar equation to calculate the moment \(\vec{F}\) creates about A is:
Notice that the magnitude of a moment depends only on the magnitude of the force and the length of the moment arm. Every point on the wrench experiences a different moment due to the force, with the points on the force's line of action having no moment due to the zero-length moment arm from these points. The direction of the moment at each point is determined by the right-hand rule as outlined in the previous section. If the bolt at A is difficult to remove, we can apply a larger moment. This could be done by 1) optimizing the direction of the force relative to the wrench, 2) increasing the magnitude of the force, or 3) moving the force further from the axis of rotation.
This interactive adds moment arms to the wrench example outlined in Figure 4.1.2.
Notice that the moment arm always points FROM the point you are taking the moment about TO the force’s line of action. Practice using the right-hand rule to determine the direction of the moment as you rotate the wrench and \(\vec{F}\text{.}\) We will introduce the use of the general position vector \(\vec{r}\) in the sections ahead.
Varignon’s Theorem is a mathematical simplification of taking cross products developed by French mathematician Pierre Varignon (1654 – 1722) which was published in 1687. It states that the sum of the moments of multiple concurrent forces is equal to the moment of the resultant of those forces. In other words, to find the moment of a series of concurrent forces about a given point, we can sum the forces first to find the resultant and then calculate the moment of that resultant about the point of interest, or compute the moment of each individual force about the point of interest and then sum those moments to find the net moment about that point.
Either method will give us equivalent results.
This can be a convenient tool if you know the Cartesian components of a force (\(F_x\) and \(F_y\)) and want to calculate the moment of that force about a point. Varignon’s Theorem tells you that you can calculate the moment of \(F_x\) and \(F_y\) about that point individually, and then add the results to find the net moment about the point. As an added bonus, the \(x\) and \(y\) components of the position vector and the force vector are naturally perpendicular, so you do not need to apply the full cross product. Simply multiply the perpendicular components and assign signs for each moment from the right-hand rule. In the case of a two-dimensional moment this would look like: