College of Engineering | University of Nebraska–Lincoln



Moment of Force about a Point

Definition: [pic], where [pic]is a position vector from point ������ to any point on the line of action of force [pic] (sliding vector).

Physical meaning: Measure of rotational tendency of [pic] with respect to ������.

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: ������=������. :

[pic]

Vector definition:

To find the moment of [pic] about an axis ������−������:

1) Select an arbitrary convenient point ������ on the axis ������−������

2) Compute moment of [pic] about the point ������

[pic]

where ������ is an arbitrary convenient point on line of action of [pic]

3) Compute projection of [pic] onto ������−������:

[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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