Section 4.4 Moment about a Point
Key Questions
At the end of this chapter you should be able to answer these questions.
- Where does the moment arm vector \vec{r} start and end?
- Why do we often use determinants to find three-dimensional moments?
Moments exist in three-dimensional systems just like those in two-dimensional systems. The main challenge when getting these is that moments now can exist in all three component directions, as opposed to only around the Z-axis. two-dimensional moments we expressed with rotational arrows as we were looking straight at the ±Z-axis of rotation. Three-dimensional moments are shown in this book as a red double-headed arrow and can be drawn in vector space just like a force or position vector.
When calculating a moment about a point in three-dimensions, it is usually not convenient to find the length of the moment arm perpendicular to the Three-dimensional force. We will instead focus on the use of the cross-product determinant, which automatically applies the right-hand rule to all component combinations.
Recall that the position vector \(\vec{r}\) goes from the point you are taking the moment about to the line of action of the force. Figure 4.4.1 shows what the computations and vectors look like when taking a moment about the origin of the axis system.
Interactive showing how crossing the position vector \(\vec{r}\) into force \(\vec{F}\) results in a moment \(\vec{M}\) which is perpendicular to both \(\vec{r}\) and \(\vec{F}\) with its direction determined from the right-hand rule.
In Three-dimensional, the matrix form is:
As before, \(r_x\text{,}\) \(r_y\text{,}\) and \(r_z\) are components of the vector describing the distance from the point of interest to the force. \(F_x\text{,}\) \(F_y\text{,}\) and \(F_z\) are components of the force. Unlike in two-dimensional, the result of the cross-product can have three components. These components represent the moments around each of the \(x\text{,}\) \(y\text{,}\) and \(z\) axes. The magnitude of the moment can be calculated in a similar way to that used to calculate the magnitude of a force vector.
\left |\vec{M} \right | = \sqrt{ {M_x}^2 +{M_y}^2+{M_z}^2 }
Figure 4.4.2 below provides a more flexible interactive than Figure 4.4.1 above as the point you are taking a moment about can vary in location. Additionally the position vector \(\vec{r}\)
Interactive showing a Three-dimensional moment M about the variable location point C. Recall that the definition of a moment tells us that \(\vec{M}\) is the magnitude and axis around which \(\vec{F}\) causes rotation] about point C.