CE 201 - Statics Lecture 11. MOMENT OF A FORCE-VECTOR FORMULATION In scalar formulation, we have...
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Transcript of CE 201 - Statics Lecture 11. MOMENT OF A FORCE-VECTOR FORMULATION In scalar formulation, we have...
CE 201 - Statics
Lecture 11
MOMENT OF A FORCE-VECTOR FORMULATION
In scalar formulation, we have seen that the moment of a force F about point O or a bout a moment axis passing through point O is equal to:
+ MO = F dwhere d is the perpendicular distance between the force and point O.In vector form, the moment can be expressed as:
+ MO = r Fwhere r is the position vector drawn from O to any point on the line of action of F. The moment has both magnitude and direction.
Magnitude+ MO = r F = r F sin = F (r sin ) = F d
A
F
rO
MO
Moment Axis
A
F
rO
MO
Moment Axis
d
Transmissibility of a Force
F has a moment about point O equals to:
MO = r F
We have seen that r can be extended from point O to any point on the line of action of force F. Then:
MO = rA F = rB F = rC F
Then F has the property of a sliding vector and, therefore can act at any point along its line of action. F is therefore a transmissible force.
A
F
rA
O
MOrB
B
C
rC
x
y
z
Cartesian Vector Formulation
If F and r are expressed as Cartesian vectors, then
i j k
+ MO = r F = rx ry rz
Fx Fy Fz
If the determinant was determined, then,
+MO = (ryFz - rzFy) i – (rxFz – rzFx) j + (rxFy – ryFx) k
The vector formulation method has the benefit of solving three-dimensional problem easier. Position vectors can be easily formed rather than perpendicular distance ( d ).
F
r
O
MO
MomentAxis
x
y
z
z
y
x
r
rx
ry
rz
F
Fx
Fy
Fz
Resultant Moment of a System of Forces
+ MRO = ( r F )
F1
r1
F2
F3x
y
z
r2
r3
Principle of Moments
If F = F1 + F2+ MO = ( r F1 ) + ( r F2 ) = r (F1 + F2 ) = r F
The principle of moments states that “the moment of a force about a point is equal to the sum of the moments of the force’s components about the point”.
F
F1
F2r
O