College of Engineering | University of Nebraska–Lincoln

Moment of Force about a Point
Definition: [pic], where [pic]is a position vector from point í µí±¶ to any point on the line of action of force [pic] (sliding vector).
Physical meaning: Measure of rotational tendency of [pic] with respect to í µí±¶.
Units: S.I. – N * m, U.S.C.S. – lb * ft
Question:
Where the moment points out?
Varignon’s theorem – moments are distributive
The moment about a given point O of the resultant of several concurrent forces is equal to the sum of the moments of the various forces about the same point O.
[pic]
2D Moments
Moment of plane force about a point in the same plane always points out-of-plane. It can, therefore, be considered a scalar.
Since for any [pic] and [pic], [pic]:
[pic]
Sign convention: counter clock-wise is positive (out of the paper plane), clock-wise is negative (into the paper plane).
Computation of 2D Moments
1st method: Use scalar definition [pic], by finding d and F=[pic]
2nd method: Resolve [pic] into components and use Varignon’s theorem.
Note 1: components don’t have to be rectangular (but should add-up to be equal to the original vector).
Note 2: signs/directions of those moments from components can be opposite!
3rd method: Use vector definition [pic] and perform multiplication algebraically.
[pic]
Note: In 2D [pic] have zero z-components, while [pic] has only z-component.
Example 1
Example
[pic]
[pic]
Example 3
[pic]
Example 4
[pic]
Example 5
[pic]
[pic]
Example 6
[pic]
[pic]
Example 7
[pic]
[pic]
3D Moments
Unlike 2D moments, in 3D all vectors in [pic] can have all three components simultaneously non-zero.
Computation of 3D moments done almost exclusively by vector algebra: cross product of [pic] and [pic]. Occasionally the second method (Varignon’s theorem) is used.
Similarly to 2D, the distance from O to line can be computed using moment formulas:
[pic]
Moment (3D), Example 1
[pic]
[pic]
Example 2
[pic]
[pic]
Example 3:
[pic]
Moment about an axis
In some cases rotation a body can be restricted to occur about an axis. In this case a new useful mechanical quantity can be derived – moment about an axis.
[pic]
To compute this moment:
* Compute a moment [pic]about any point on the axis,
* Compute a projection of [pic] onto the axis.
Example from the above figure:
1) [pic]
2) [pic]
Alternatively, find the distance from the line of action to the axis: í µí²…=í µí¿Ž. :
[pic]
Vector definition:
To find the moment of [pic] about an axis í µí²‚âˆ’í µí²‚:
1) Select an arbitrary convenient point í µí±¶ on the axis í µí²‚âˆ’í µí²‚
2) Compute moment of [pic] about the point í µí±¶
[pic]
where í µí±¨ is an arbitrary convenient point on line of action of [pic]
3) Compute projection of [pic] onto í µí²‚âˆ’í µí²‚:
[pic]
4) Define the vector [pic]
[pic]
Note: from the above definitions, we can obtain a physical meaning of determinant:
Components of moment vector: Mx, My, Mz are moments of force [pic] about the corresponding axes passing through O!
Moment about an axis, Example 1
[pic]
Example 2:
[pic]
[pic]
Example 3:
[pic]
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