RIGID BODIES: MOMENTS OF
FORCES AND COUPLES
ES 11 Statics of Rigid Bodies
Lecture 4
RIGID BODY
Treatment of a body as a single particle is not always possible.
In general, the size of the body and the specific points of
application of the forces must be considered.
Forces will act on different particles and thus will have different
points of application (LOA).
Most bodies in elementary mechanics are assumed to be rigid,
i.e., the actual deformations are small and do not affect the
conditions of equilibrium or motion of the body.
RIGID BODY
a body which does not deform under the
action of forces.
an idealization of a solid body of finite size in
which deformation is neglected
the distance between any two given points of
a rigid body remains constant regardless of
external forces exerted on it.
PRINCIPLE OF TRANSMISSIBILITY:
EQUIVALENT FORCES
Principle of Transmissibility states that the conditions
of equilibrium or motion of a rigid body will remain
unchanged if a force acting at a given point of the rigid
body is replaced by a force of the same magnitude and
direction, but acting at a different point, provided that the
two forces have the same line of action (LOA).
MOMENT OF A FORCE
Moment of a force is the
measure of the tendency of a
force F to make the rigid
body rotate about a fixed axis
perpendicular to the plane of
the force F.
Mo
F
r
d
A
pt. O fixed point/axis on the plane of the force and pt.
r - position vector of F acting at pt. A relative to the fixed pt. o.
pt. A pt. of application of force F
MOMENT OF A FORCE ABOUT A POINT
Mo
F
(vector)
d
A
(scalar or magnitude)
where d represents the perpendicular distance form O to the line of
action of F.
MOMENT OF A FORCE ABOUT A POINT
EXAMPLE:
A 150-N force is applied to
the control rod AO as
shown. Knowing that the
length of the rod is 350 mm,
compute the moment of the
force about point O using
= x .
MOMENT OF A FORCE ABOUT A POINT
SOLUTION:
= <-350cos70, 350sin70, 0> X
<15cos35, -150 sin 35, 0>
= <0, 0, -30112.76> N-mm
RECTANGULAR COMPONENTS OF THE
MOMENT OF A FORCE
VARIGNONS THEOREM
The moment about a
given point O of the
re s u l t a n t o f s e ve ra l
concurrent forces is
equal to the sum of the
moments of the various
moments about the
same point O
MOMENT OF A FORCE ABOUT A POINT
EXAMPLE:
Forces P and Q of
magnitude 500 N and
300 N, respectively,
are applied on the
bent board as shown.
Determine the
moment of these two
forces about point D.
MOMENT OF A FORCE ABOUT A POINT
SOLUTION:
MOMENT OF A FORCE ABOUT A POINT
SOLUTION:
MOMENT OF A FORCE ABOUT A LINE
Consider a line L and a force F. Let MP be the moment
of F about an arbitrary point P on L.
Mp
M p = rx R
P
ML
MOMENT OF A FORCE ABOUT A LINE
EXAMPLE:
Forces P and Q of
magnitude 500 N and
300 N, respectively,
are applied on the
bent board as shown.
Determine the
moment of these two
forces about the line
passing through points
O and D.
MOMENT OF A FORCE ABOUT A LINE
SOLUTION:
COUPLES
A pair of forces of:
Equal magnitudes
Opposite directions
Parallel LOAs
A couple tends to cause
rotation of an object even
though the vector sum of the
forces is zero.
COUPLES
Why are these forces non-couples?
EXTERNAL EFFECT OF A COUPLE
A couple can only
cause rotation of an
object.
The resultant force of
a couple is zero.
Its resultant moment
is not zero.
MOMENT OF A COUPLE
MOMENT OF A COUPLE
Moment of a couple is constant.
MOMENT OF A COUPLE
MOMENT OF A COUPLE
EXAMPLE:
The force F is 10i
4j (N). Determine
the moment of the
couple.
MOMENT OF A COUPLE
SOLUTION:
COUPLE MAGNITUDE : Fd
From the definition of the moment of a couple, it follows
that two couples, one consisting of the forces F1 and F2,
will have equal moments if
TRANSFORMATION OF A COUPLE
The magnitude of the forces that make up the couple can be changed with
an opposite change in the perpendicular distance without changing the
effect (magnitude) of the couple.
10N
20N
1.0m
10N
0.5m
20N
TRANSFORMATION OF A COUPLE
If you can't turn the nut of
a wheel, what do you do to
turn it?
You can call a stronger
person to turn it, thereby
increasing F, or
You can place your
hands farther apart on the
wrench, thereby
increasing d.
TRANSFORMATION OF A COUPLE
The directions of the forces can be changed as long as
the magnitude of the couple and the orientation of the
plane where the forces are acting remains the same.
10N
10N
1.0m
10N
1.0m
10N
TRANSFORMATION OF A COUPLE
Two couples having the same moment M, are
equivalent even if they are at different planes as long
as those 2 planes are parallel.
COUPLE VECTORS
ADDITION OF COUPLE VECTORS
Couple vectors obey the
law of vector addition.
The couple vector M may
be resolved into
component vectors Mx
My , and Mz directed along
the axes of coordinates
and representing couples
acting, respectively in the
yz, xz, and xy planes.
ADDITION OF COUPLE VECTORS
EXAMPLE:
Replace the three
couples shown
with a single
equivalent couple
M, specifying its
magnitude and
direction.
ADDITION OF COUPLE VECTORS
SOLUTION:
ADDITION OF COUPLE VECTORS
SOLUTION:
