Download Moment and Torque

Moment and
Torque
Outline
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Ozkaya and Nordin, Ch. 3 (p. 31-46)
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Moments cause twisting or bending
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Moment is force times distance
effect of a force F on a rigid body depends on
where it is applied
torque (T) or moment (M): measure of the
tendency of a force to cause rotation or bending
about a specific axis
units of N·m
M
F
definition of moment and moment arm
conventions for defining moment direction
addition of moments, moments due to force
components
moments due to external forces
moments due to muscles
translation of forces and equivalent systems
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the moment that a force produces about a fixed
point O is equal to the force times the
perpendicular distance from O to the line of
action of the force
d2
d1
F = mg
r
v
MO = Fd !
d ! = "moment arm"
r
F
(M )
O 2
T
O
F1 = 100 N
F2 = 100 N
r
> ( MO )1
Moment arm is perpendicular to line of
action of force
Make sure you
calculate moment based
on the moment arm, not
the distance from the
point of force
application to the axis
of rotation.
Moment is a Vector
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F1
d1
F2
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d3
F3
P
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a moment has both magnitude
and direction
in static equilibrium problems,
there is no strict convention for
defining positive and negative
moments
we will usually refer to a CCW
moments as positive, and a CW
moments as negative
Right Hand Rule
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moments can be represented in
diagrams one of two ways
the direction of MO is
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perpendicular to the plane formed
by F and O
determined by the “right hand rule”
right hand rule: identify the direction
that the moment tends to cause the
body to rotate about point O
orient your right hand so this
direction is illustrated by the act of
curling your fingers into increased
flexion
your thumb will be pointing in the
direction of the moment
Moment addition
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representation that
shows the direction
of MO
r
F
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Moments about a given axis add algebraically
Be careful to account for the sign (i.e.,
direction) of each moment
r
M
+ MO = !2 " W2 ! 4 " W1
Moment calculations can be simplified by
using force components
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It is often convenient
to resolve a force F
into its x, y, and z
components in order
to calculate the
moment it generates
at a specific location.
External Forces Create Moments
The effect of external loads
on producing human body
movements, and generating
internal forces and stresses,
depends on the moments
they create.
Consider, for example, how
the angle ! influencces the
feasibility or effort involved
in the bench press exercise.
+ MB = Fy (x A ! x B ) ! Fx (yA ! yB )
Muscles Create Moments
The ability of muscles to
stabilize the body under
external loads, and produce
human body movements, is
due their ability to create
moments (or torques) about
joints.
If the line of action of a
muscle force passes through
a joint axis of rotation, it
cannot stabilize or rotate that
joint.
Muscles have small moment arms
Compared to external loads, muscles have
small moment arms with respect to joint
they span.
Advantage: given the limited deformation
that muscles can undergo, their close
proximity to joints allows for large ranges
of joint rotation
Disadvantage: high muscle forces (and
therefore joint forces) are required to
balance external loads
Forces can be moved by replacing
them with a force and a moment
Elbow angle effects moment arm of
elbow flexor muscles
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Moment Arm (mm)
100
Brachialis
75
Biceps
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A force F applied to a rigid body at a location A can be
translated to another location O on the body, if a moment
(or couple) MO is also added at O.
The location of O will affect the magnitude of MO.
Brachioradialis
50
2D
25
0
50
100
3D
150
Elbow Angle (degrees)
Multiple forces can be replaced with
a force and a moment
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Multiple forces applied to a rigid body can be
replaced by a single resultant force and a single
moment, acting at any location on the body.
The location will affect the resultant moment, but not
the resultant force.
Review Questions
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Multiple
external loads
Equivalent
resultant load
and moment
Equivalent
single load
How is a moment defined (mathematically)?
How is a moment arm defined?
How is the right hand rule used to determine
the direction of a moment?
How do moments add?
How can several forces be replaced with a
single moment and force?
How does elbow rotation affect the moment
arm of elbow flexor muscles? Does this
explain why elbow strength varies with elbow
angle?
Tutorial problems:
1. Determine the magnitude of
Mx, My, and Mz at the origin
O, in terms of force
components Fx, Fy, and Fz,
and distances xA, yA, and zA.
2. Determine moments at the
elbow (point B) and shoulder
(point A) due to the vertical
load F.