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3 Moment of Force about a PointDefinition: M o=r× F, where ris a position vector from point 𝑶 to any point on the line of action of force F (sliding vector).

Physical meaning: Measure of rotational tendency of F with respect to 𝑶.

Units: S.I. – N * m, U.S.C.S. – lb * ft

Question: Where the moment points out?

Varignon’s theorem – moments are distributiveThe 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.

r × (F1+F2+… )=r ×F1+r ×F2+…

2D MomentsMoment 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 r and θ, r∗sin (θ )=d:M o=|r× F|=|r|∗|F|∗sin (θ )=d∗F

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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 M o=d∗F, by finding d and F=|F|

2nd method: Resolve F 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 M o=r× F and perform multiplication algebraically.

M o=|r× F|=r∗F∗sin (θ )=d∗F

Note: In 2D r∧F have zero z-components, while M o has only z-component.

Example 1

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Example

Example 3

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Example 4

Example 5

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Example 6

Example 7

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3D MomentsUnlike 2D moments, in 3D all vectors in M o=r× F can have all three components simultaneously non-zero.

Computation of 3D moments done almost exclusively by vector algebra: cross product of r=rx i+r y j+r z k and F=F x i+F y j+F z k. Occasionally the second method (Varignon’s theorem) is used.

Similarly to 2D, the distance from O to line can be computed using moment formulas:

d∗F=|r ×F|

Moment (3D), Example 1

Example 2

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Example 3:

Moment about an axis

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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.

To compute this moment:• Compute a moment M oabout any point on the axis,• Compute a projection of M o onto the axis.

Example from the above figure:1) |M o|=20 ∙0.5=10N ∙m2) M y=10 ∙35=6N ∙m

Alternatively, find the distance from the line of action to the axis: 𝒅=𝟎. :

⇒M y=20 ∙0.3=6N ∙m

Vector definition:To find the moment of F about an axis 𝒂−𝒂:1) Select an arbitrary convenient point 𝑶 on the axis 𝒂−𝒂

2) Compute moment of F about the point 𝑶M o=rOA×Fwhere 𝑨 is an arbitrary convenient point on line of

action of F

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3) Compute projection of M o onto 𝒂−𝒂:

M aa=M o ∙uaa=uaa ∙ ( rOA×F )=|ux u y uzrx r y rzFx F y F z|

4) Define the vector M aa M aa=M aa ∙ uaa=(uaa ∙ (rOA×F ))∙ uaa

Note: from the above definitions, we can obtain a physical meaning of determinant:

Components of moment vector: Mx, My, Mz are moments of force F about the corresponding axes passing through O!

Moment about an axis, Example 1

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Example 2:

Example 3:

A