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Engineering Statics: Open and Interactive

Daniel W. Baker, William Haynes

Section 4.1 Direction of a Moment

In a two-dimensional problem the direction of a moment can be determined easily by inspection as either clockwise or counter-clockwise. A counter-clockwise rotation corresponds with a moment vector pointing out of page and is considered positive.
In three-dimensions a moment vector may point in any direction in space and is more difficult to visualize. The direction is established by the right hand rule.
To apply the right hand rule, first establish a position vector \(\vec{r}\) pointing from the rotation center to the point of application of the force, or another point on its line of action. If you align your thumb with the position vector and your index finger with the force vector, then your middle finger points the direction of the moment vector \(\vec{M}\text{.}\) Alternately, you can align your index finger with the position vector and your middle finger with the force vector, and your thumb will point in the direction of the moment vector.
Using the right hand, point index finger straight out, the thumb up (like giving a thumbs-up), and the middle finger pointed perpendicular to the pointer finger and thumb. The thumb represents the position vector, and the index finger represents the force vector. Taking the cross product r x F results in M which is represented by the middle finger. Alternatively, r can be the index finger, F the middle finger, and M the thumb.
Figure 4.1.1. Two ways to apply the right hand rule to determine the direction of a moment.
Another approach is the point-and-curl method. Start with your right hand flat and fingertips pointing along the position vector \(\vec{r}\) pointing from the center of rotation to a point on the force’s line of action. Rotate your hand until the force \(\vec{F}\) is perpendicular to your fingers and imagine that it pushes your fingers into a curl around your thumb. In this position, your thumb defines the axis of rotation, and points in the direction of the moment \(\vec{M}\text{.}\)
Image of a right hand held out flat, where \(\vec{r}\) is in the direction of the fingers held out flat. If the fingers were curled in, and the thumb pointed up, then the curled fingers are in the direction of \(\vec{F}\) and the thumb in the direction of \(\vec{M}\text{.}\)
Figure 4.1.2. Point-and-curl right-hand rule technique for moments.
Consider the page shown below on a horizontal surface. Using these techniques, we see that a counter-clockwise moment vector points up, or out of the page, while the clockwise moment points down or in to the page. In other words, the counter-clockwise moment acts in the positive \(z\) direction and the clockwise moment acts in the \(-z\) direction.
A counter-clockwise moment symbol on a horizontal page, and the corresponding unit vector pointing up, also a similar clockwise moment pointing down.
Figure 4.1.3. Moments in the plane of the page.
Any of these techniques may be used to find the direction of a moment. They all produce the same result so you don’t need to learn them all, but make sure you have at least one method you can use accurately and consistently.
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