EGR 280
Mechanics
2 – Moments, equivalent systems of forces
Statics of Rigid Bodies
Principle of Transmissibility: Forces can be moved along their lines of
action without changing the overall effect on any rigid body. Any
deformations or local changes in the body are, of course, dependent on where
the forces are applied.
F
F
Moment of a force about a point: Given a point O in space and a force F, the
moment of F about O is defined as:
MO = r × F
where r is the position vector
from O to anywhere on the
line of action of F
MO = rFsinθ = Fd
where d = rsinθ
F
θ
r
O
Rectangular components:
i j k
(ryFz – rzFy) i
MO = r × F = r x r y r z =
+ (rzFx – rxFz) j
Fx Fy Fz
+ (rxFy – ryFx) k
d
Moment of a force about a given axis:
F
O
λL
rO
Given a point O, force F and an axis defined by the unit vector λL, the
magnitude of the component of the moment of F about O in the λL direction is:
MOL = (rO × F) • λL
And the moment of F about axis is
MOL = [(rO × F) • λL)] λL
Note: rO is from anywhere on axis λL to
anywhere on the line of action of F
The magnitude of a moment is a measure of the tendency of the force to rotate
the rigid body about point O. The direction of the moment give the direction
of the rotation, by the right-hand rule.
F
rO
MO = rO × F
O
Moment of a couple: Two forces F and –F which have the same magnitude,
parallel lines of action and opposite sense are said to form a couple.
|M| = |r×F|
= rFsinθ
= Fd
d
F
-F
r
Equivalent couples – couples that have the same magnitude and sense are
equivalent.
Since couples are moments, they can be written as vectors and all vector
operations applied to them.
We have seen that we can move forces freely along their lines of action.
Moving a force parallel to its line of action produces a couple:
M
-F
O
F
O
O
F
F
F
Equivalent systems of forces: Two force systems are equivalent if they can be
reduced to the same force-couple system at any point.