Download Moment of a Couple

Forces and Moments
Lecture 3
What is a moment?
 A moment is a type of force.
 A moment is a turning force.
 A turning force happens when a force is applied to
something which has a pivot.
 The moment of a force: with respect to a line perpendicular to a plane containing the
force is: the product of the force and the perpendicular distance from the force to the line or
moment axis.
Mo = F x d
F
90o
d
Moment of Force F around point O
Lecture 3
 The moment of a force about a point or an
axis (MO), provides a measure of the tendency
of the force to cause a body to rotate about the
point or axis.
 Oz ┴ plane xy in which Fx lies.
 Fx causes the pipe to turn about the z-axis.
 Fx causes a moment about the z-axis = (Mo)z
NO moment!!
 Fy passes through O.
 Fy does not cause the pipe to turn because
the line of action of the force passes through
O.
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Moment of Force F around point O




Lecture 3
Ox ┴ plane zy in which Fz lies.
Fz causes the pipe to turn about the x-axis.
Fz causes a moment about the x-axis (Mo)x .
Fz is parallel to z- axis, the moment of Fz
about z-axis is equal to zero. Fz tends to
translate the pipe in z - direction
3 - D Moment
If the force does not lie in a plane perpendicular to the moment axis, it may be resolved into
two or three, one of them being parallel to the moment axis and the others lying in a plane
perpendicular to the axis of moment or rotation.
Z
A
Z
B
C
C
F
Fz
B
F
D
D
Fz
G
F
X
H
Fx
G
F
X
E
E
Y
A
Y
Fy
Fx
H
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Lecture 3
Moment (Scalar or Vector?!)
 Moment is a vector and the direction is defined using right hand rule. Keep
your fingers along the line of action of the force, and your thumb will be along the
axis of the moment.
M
M
F
r
M
Magnitude of the Moment
M
F
Direction of Moment
The direction of Mo would be specified by using the right hand rule.
Counter Clockwise (CCW) is out of the page
Clockwise (CW) is into the page.
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Calculating the moment in 2-D using components



Lecture 3
Select a positive direction (CCW or CW).
Calculate each moment and add them, using the proper sign for each term.
Always remember to write the unit of moment which is (N.m or lb.f).
Example 1: In the following figure, calculate the moment about the point O.
1. We choose the CCW as positive direction for moment,
2. Moment of component of F along x about O is Fx times the perpendicular distance
from O (or d1), which is clockwise, so it is ( - Fx . d1 )
3. Moment of component of F along y about O is Fy times the perpendicular distance
from O (or d2), which is counter clockwise, so it is (Fy . d2 )
4. Moments add together as vectors, so the total moment is:
F
M o   Fx d1  Fy d 2
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Lecture 3
Calculating the moment in 2-D using components
Example 2: In the following figure, if ( q ) is 60 degrees and r is 30 mm and F is 6 N,
what is the magnitude of the moment about O.
1. We choose the CCW as positive direction for moment,
2. Component of F along r (or F1 = F cos q ) produces no moment, since it passes from
point O.
3. Component of F perpendicular to r (or F2 = F sine q ) produces the moment.
So the total moment of F about O is:
M O  F r   F sin q  r  6  sin 60  .03  .156 Nm
Remember:
The moment about O is also calculated using
the magnitude of force F times perpendicular
distance from O to the line of action of F which
is d :
M O  Fd  F  r sin q   6  .03  sin 60  .156 Nm
F2
F
q
q
F2
F1
Note: Moving a force along its line of action does not change its moment!
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Lecture 3
Moment of a Couple
A couple is defined as:
Two parallel forces [ F and ( – F ) ] have Same
magnitude, Opposite direction and separated by a
perpendicular distance ( d ).
-F
d
F
Couple Moment
A moment produced by a couple is called a Couple Moment.
But:
rB - rA = r
(as a vector)
Thus:
M = r x F
So a couple moment is a free vector which can act at any point and depends
only on r, not on rA and rB.
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Remember:
Scalar Formulation:
Moment of a Couple
Lecture 3
Magnitude:
M=F.d
Direction and sense using right-hand rule.
Vector Formulation:
Magnitude:
M=dxF
Note: The moment of a couple does not depend on the point one takes
the moment about. In other words, a moment of a couple is the same
about all points in space.
Example 3:
The crossbar wrench is used to remove a lug nut from
the automobile wheel. The mechanic applies a moment
couple to the wrench such that his hands are a constant
distance apart.
 Is it necessary that a = b in order to produce the most
effective turning of the nut? Explain.
 Also what is the effect of changing the shaft
dimension c in this regard? The forces act in the
vertical plane.
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Lecture 3
Moment of a Couple
Example 3:
Solution:
 Couple moment:
Mcouple = F . ( a + b ),
The couple moment depends on the total distance between grips. a = b is not a necessary
condition to produce the most effective turning of the nut.
 Changing the dimension c has no effect on turning the nut.
Example 4:
Determine the couple moment of the two couples that act on the pipe assembly about ( X, Y
and Z) axes. The distance from A to B is d = 400 mm. Express the result as a Cartesian
vector.
50 N
- 50 N
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Moment of a Couple
Lecture 3
Example 3:
Solution:
• Lets check three different views and see what forces are causing moments about x, y and z axes:
• Scalar analysis: Summing moments about ( X, Y and Z ) axes.
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