3.50
3 Moment of Force about a Point
Definition:
, where is a position vector from point
𝑶 to any point on the line of action of force (sliding vector).
Physical meaning: Measure of rotational tendency of
respect to 𝑶.
Units:
S.I. – N * m,
with
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.
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 and ,
:
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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
F=
2nd method: Resolve
theorem.
, by finding d and
into components and use Varignon’s
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
multiplication algebraically.
and perform
Note: In 2D
have zero z-components, while
only z-component.
has
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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 Moments
Unlike 2D moments, in 3D all vectors in
all three components simultaneously non-zero.
can have
Computation of 3D moments done almost exclusively by vector
algebra: cross product of
and
. Occasionally the second method
(Varignon’s theorem) is used.
Similarly to 2D, the distance from O to line can be computed
using moment formulas:
Moment (3D), Example 1
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Example 2
Example 3:
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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.
To compute this moment:
•
•
Compute a moment
about any point on the axis,
Compute a projection of
onto the axis.
Example from the above figure:
1)
2)
Alternatively, find the distance from the line of action to the
axis: 𝒅=𝟎. :
Vector definition:
To find the moment of about an axis 𝒂−𝒂:
1) Select an arbitrary convenient point 𝑶 on the axis 𝒂−𝒂
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2) Compute moment of
about the point 𝑶
where 𝑨 is an arbitrary convenient point on line of action of
3) Compute projection of
onto 𝒂−𝒂:
4) Define the vector
Note: from the above definitions, we can obtain a physical
meaning of determinant:
Components of moment vector: Mx,
My, Mz are moments of force about
the corresponding axes passing
through O!
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Moment about an axis, Example 1
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Example 2:
Example 3:
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