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Chapter 5
Torques and Moments of Force
Maintaining Equilibrium or Changing
Angular Motion
Objectives
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Define torque
Define static equilibrium
List the equations of static equilibrium
Determine the resultant of two or more
torques
• Determine if an object is in static
equilibrium, when the forces and torques
acting on the object are known
Objectives
• Determine an unknown force (or torque)
acting on an object, if all the other forces
and torques acting on the object are
known and the object is in static
equilibrium
• Define center of gravity
• Estimate the location of the center of
gravity of an object or body
What Are Torques?
• Turning effect produced by a force is called a
torque
• May also be called a moment of force or
moment
• External force directed through COG of an
object is called a centric force—Causes a
change in the linear motion of an object
• External force not directed through the COG of
an object is called an eccentric force (type of
force not type of muscle action in this case)—
Causes a change in the linear and angular
motions of an object
What Are Torques?
• Pair of external forces acting in equal but
opposite directions is called a force
couple—Causes a change only in the
angular motion of an object
• Resultant of the two forces in a force
couple is zero
Mathematical Definition of Torque
• Torque produced by a force directly
proportional to the size of the force and
the distance between the line of action of
the force and the point about which the
object tends to rotate
• Moment arm—Perpendicular distance
between the line of action of the force and
a line parallel to it that passes through the
axis of rotation
Mathematical Definition of Torque
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Torque is defined mathematically as:
T = Fr
T = torque (or moment of force)
F = Force (Newtons)
r = moment arm (meters)
Mathematical Definition of Torque
• Vector quantity—Turning effect is around a
specific axis that is directed in a specific
direction
• Counterclockwise torques are positive
• Clockwise torques are negative
• Torques acting about the same axis may
be added or subtracted to determine the
resultant
Examples of How
Torques Are Used
• Why do you suppose doorknobs or door handles
are located on the opposite side of the door from
the hinges?
• Same size torque can be created with a large
force and a small moment arm or with a small
force and a large moment arm
• Because the amount of force humans can exert
is generally limited, we use large moment arms
when we want to create large torques
Examples of How
Torques Are Used
• How do common tools we use increase
torque?
• Other everyday objects?
• Why do heavy trucks have larger-diameter
steering wheels than cars?
• How is torque used in sport?
• In any sport in which we turn, spin, or
swing something (including our bodies),
torque must be created
Muscular Torque
• What about torques within the body?
• Muscles create torques that turn our limbs
• Line of action of a muscle force is some
distance from the joint axis
• Torque produced the muscle on the distal
limb will tend to rotate that limb in one
direction about an axis through the joint
Muscular Torque
• What happens to the torque on the
forearm produced by the biceps brachii
muscle as the forearm is moved from full
extension to 90° of flexion at the elbow
joint?
• Can the muscle create the same torque
throughout this range of motion?
Muscular Torque
• Changing the angle at the joint changes
the moment arm of the muscles that cross
that joint—Partially explains why our
muscles are apparently stronger in some
joint positions than others
Strength-Training Devices
and Torque
• What happens to the torque produced around
the elbow joint by the dumbbell when an arm
curl exercise is performed?
• Dumbbell doesn’t get heavier, but the torque
gets larger up to 90 degrees of elbow flexion
• Most free weight exercises, torques produced by
the weights vary as the moment arms of these
weights change during the movement
Strength-Training Devices
and Torque
• With weightlifting machines, cables or
chains are used to redirect the line of
action of the force of gravity acting on the
weight stack
• Nautilus weightlifting machines are
designed so that the resistive torque
varies in proportion to the changes in the
moment arm of the muscle being
exercised
Forces and Torques in Equilibrium
• For an object to be in static equilibrium,
the external forces and torques acting on it
must sum to zero
• Sample Problem 5.1 (p. 126 text)
Net Torque
• Torques that act around the same axis can be
added or subtracted algebraically
• Net torque is computed by summing the torques
that act on an object
• Example
– Pennies placed to left of eraser cause rotation in
counterclockwise direction (positive torque)
– Pennies placed to right of eraser cause rotation in
clockwise direction (negative torque)
– How can we achieve static equilibrium?
Muscle Force Estimates Using
Equilibrium Equations
• How much torque is created about the
elbow joint axis while holding a 20 lb
dumbbell with the elbow joint flexed at 90°
if the length of the forearm is 12 in?
–T=Fxr
• What force must the muscles produce to
generate sufficient torque to hold the
dumbbell if the point of insertion is 1 in
from the elbow joint axis?
More Examples of Net Torque
• What external forces act on a polevaulter?
– Gravity pulls downward on the vaulter with a
force equal to his/her weight
– The pole exerts reactive forces on the
vaulters hands where he/she grips the pole
– What net torque acts on this vaulter around
an axis through his/her center of gravity (just
after takeoff)? Is the vaulter in equilibrium?
More Examples of Net Torque
• 500 N force acting on vaulters left hand has a
moment arm of .5 m about his/her center of
gravity—creates clockwise torque
• 1500 N force acting on the vaulters right hand
has a moment arm of 1.0 m about his/her center
of gravity—also clockwise
• Vaulters weight of 700 N acts through center of
gravity—moment arm is zero so zero torque
More Examples of Net Torque
• ΣT = Σ(F x r) = (-500 N)(.5 m) + (-1500
N)(1.0 m) = -1750 Nm
• Negative sign indicates clockwise direction
• Produces turning effect that ends to rotate
the vaulter onto his back (i.e. backward
somersault)
• What happens later in the vault?
More Examples of Net Torque
• 300 N force acting on vaulters left hand
has a moment arm of .5 m about his/her
center of gravity—still creates clockwise
torque
• 500 N force acting on the vaulters right
hand has a moment arm of .5 m about
his/her center of gravity—but now
counterclockwise
More Examples of Net Torque
• ΣT = Σ(F x r) = (-300 N)(.5 m) + (500 N)(.5
m) = +100 Nm
• Positive sign indicates counterclockwise
direction
• Produces turning effect that ends to rotate
the vaulter forwards (i.e. forward
somersault)
What is Center of Gravity
• Center of gravity (COG)—Point in a body
or system around which its mass or weight
is evenly distributed or balanced and
through which the force of gravity acts
• Center of mass (COM)—point in a body or
system of bodies at which the entire mass
may be assumed to be concentrated—for
bodies near the surface of the earth COG
and COM considered synonymous
Locating the Center of Gravity
of an Object
• Every object composed of smaller elemental
parts—in human body represented by limbs,
trunk, and head
• Force of gravity pulls downward on each of
these smaller elemental parts—sum or resultant
of these forces represents total weight of the
object
• Force of gravity acts through a point at which the
torques produced by each of these smaller
elemental parts sums to zero
Locating the Center of Gravity
of an Object
• If an elemental part of an object moves or
changes position, the COG moves in that same
direction (e.g. raising the arms overhead raises
COG)
• If an elemental part of an object is removed, the
COG moves away from the point of removal
• If mass is added to an object, the center of
gravity moves toward the location of the added
mass
Mathematical Determination of the
COG Location
• If the weights and locations of the
elemental parts that make up an object are
known, the COG location can be
computed mathematically
• Example:
– A ruler with six pennies distributed at 2 in.
intervals is equivalent to a ruler with six
pennies stacked on it at one location, if that
location is the COG of the first ruler
Mathematical Determination of the
COG Location
• If you closed your eyes and picked up both
rulers by the end, they would feel
identical—both rulers create the same
torque about the end of the ruler
• The sum of the torques created by each of
the elemental weights (first ruler) equals
the torque created by the total weight
stacked at the center of gravity location
(second ruler)
Mathematical Determination of the
COG Location
• Mathematically, expressed as:
– ΣT = Σ(W x r) = (ΣW) x rcg
– W = weight of one element
– r = moment arm of an individual element
– ΣW = total weight of the object
– rcg = moment arm of the entire weight of the
object (location of the COG of the object
relative to the axis about which the moments
of force are being measured)
Mathematical Determination of the
COG Location
• For ruler and pennies, the COG found for
one dimension only
• For more complex objects, COG location
defined by three dimensions, because
objects occupy space in three dimensions
• Procedure repeated for each dimension
with gravity acting in a direction
perpendicular to that dimension
Mathematical Determination of the
COG Location
• Sample Problem 5.2
• A weightlifter has mistakenly placed a 20 kg
plate on one end of a barbell and a 15 kg plate
on the other end. The barbell is 2.2 m long and
has a mass of 20 kg without the plates on it.
The 20 kg plate is located 40 cm from the right
end of the barbell, and the 15 kg plate is located
40 cm from the left end of the barbell. Where is
the COG of the barbell with the weight plates on
it?
Mathematical Determination of the
COG Location
• Sum the torques of the weights about the
right end of the barbell
– ΣT = g[(20 kg)(.4 m) + (15 kg)(1.8 m) + (20
kg)(1.1 m)] = g(57 kg m)
• Equate this to the torque of the total
weight about the right end of the barbell
and solve for rcg
– ΣT = g(55 kg) rcg = g(57 kg m)
– rcg = 1.04m
Center of Gravity of the
Human Body
• Location of COG depends on the position of
limbs
• In anatomical position, COG location 1 to 2 in
below navel—55%-57% of standing height
• Reach overhead, COG will move superiorly
• Someone with long legs and muscular arms and
chest will have a higher COG versus someone
with shorter, stockier legs
Center of Gravity of the
Human Body
• Woman’s COG slightly lower than man’s
because women have larger pelvic
girdles/narrower shoulders
• Infants and children have higher COG’s
relative to their height because of relatively
large heads and short legs
Center of Gravity of the
Human Body
• Movement of any segment of the body
causes COG to shift in same direction
– How much of a shift depends on weight of
segment and distance moved (e.g. raising leg
versus raising arm)
• COG may actually lie outside the body in
some cases
Center of Gravity of the
Human Body
• Vertical jump techniques
– Jumping with one hand overhead maximizes
vertical jump height because greater distance
between the COG and the outstretched arm
– By keeping all the limb and body parts (with
the exception of the reach hand) as low as
possible relative to the COG, the distance
from the reach hand to the COG is maximized
Center of Gravity of the
Human Body
• Basketball player vs. volleyball player
• What about “hang time”?
• COG follows parabolic path, but jumper’s head
and trunk appear to be suspended at same
height during the middle stage of the leap
• During this time, the jumper’s legs and arms rise
and then fall—these movements account for the
rise and fall of the COG, so the head and trunk
do not rise appreciably
COG and Stability
• Stability—the capacity of an object to
return to equilibrium or to its original
position after being displaced
• In many sports, the athletes do not want to
be moved from a certain stance or position
– Wrestlers, football lineman, basketball players
more successful at certain skills if they adopt
stable positions
COG and Stability
• In other sports, success may be
determined by how quickly an athlete is
able to move out of a position
– Receiver of a serve in tennis or racquetball, a
sprinter, a swimmer, a downhill skier, a goalie
in soccer more successful during certain skills
if less stable
Factors Affecting Stability
• Three primary factors:
– Height of COG
– Base of support—area within the lines
connecting the outer perimeter of each of the
points of support
– Weight
Factors Affecting Stability
• Stand a book on its edge and exert a
horizontal force against it to tip it over
– If the book remains in static equilibrium, the
net force and torque acting on the book must
be zero
Factors Affecting Stability
• External forces acting on the book include:
– Book’s weight, W, acting through its COG
– Toppling force, P
– Friction force, Ff
– Reaction Force, R
• Axis through the lower left corner of the
book
Factors Affecting Stability
• Sum of the moments about the axis equals
zero
• ΣTa= 0
• 0 = (P x h) – (W x b)
• Pxh=Wxb
Factors Affecting Stability
• Terms on the left side of the equation
minimized to increase stability
– Moment arm of the toppling force, h, related
to height of COG—lower COG implies a lower
height and a shorter moment arm for the
toppling force increases stability
Factors Affecting Stability
• Terms on the right side of the equation
maximized to increase stability
– Increasing the weight will increase the stability
because the moment of force keeping the
object upright would be larger
– Increasing the moment arm of the object’s
weight will increase stability—related to the
size of the base of support—direction of
toppling force important
Factors Affecting Stability
• Stability is directional—see Figure 5.19a
and b
• An object can be more stable in one
direction than another
• It is not the size of the base of support that
affects stability, but the horizontal distance
between the line of gravity and the edge of
the base of support in the direction that the
toppling force is pushing or pulling
Stability and Potential Energy
• Concepts of work and potential energy explain
why COG height affects stability
– Figure 5.20—As long as the COG of the block is to
the left of the lower right corner, the weight creates a
righting moment of force in opposition to the toppling
moment of force created by the force P
– When the COG is moved past the supporting corner,
the moment of force created by the weight changes
direction and becomes a toppling moment that
causes the block to topple
Stability and Potential Energy
• To move the block from its stable position
to the brink of instability, the COG had to
be raised a distance, Δh—Work was
required to do this, and the potential
energy of the block increased
• What if the COG is higher or lower?
Stability and Potential Energy
• The higher the COG, the smaller the
vertical displacement, thus the smaller the
change in potential energy and the smaller
the amount of work done
• A block with a lower COG is more stable
because more work is required to topple it
• What if the moment arm is changed?
Stability and Potential Energy
• If the distance from the line of gravity to
the edge of the base of support about
which toppling will occur is increased, the
vertical displacement the COG goes
through before the object topples also
increases, so the object is more stable
Stability and Potential Energy
• Most stable stance or position minimizes
potential energy
• Positions that place the COG below the
points of support are more stable—
Gymnast hanging from horizontal bar
• When the COG lies above the base of
support, stability is maintained only as
long as the line of gravity falls within the
base of support
Center of Gravity, Stability, and
Human Movement
• Human body not rigid—COG position and base
of support can change with limb movements
• Humans can control stability by changing stance
and body position
• Example:
– Walking—Lean forward until your line of gravity falls
in front of your feet and you lose your stability—You
begin to fall forward, and you step with one foot to
catch your fall and reestablish your stability—Walking
could be describe as a series of falls an catches
Center of Gravity, Stability, and
Human Movement
• In sports, athletes may want to maximize
their stability in general or in a specific
direction, or they may want to minimize
stability (increase their mobility)
Center of Gravity, Stability, and
Human Movement
• Wrestlers crouch to lower their COG and widen
base of support by placing feet slightly wider
than shoulder-width in a square stance or
staggered stance
• When the wrestler is in a defensive position on
his belly and trying not to be turned over onto his
back he maximizes his stability by sprawling his
limbs to the sides to maximize the size of his
base of support and to lower his COG as much
as possible
Center of Gravity, Stability, and
Human Movement
• When force is expected from a specific
direction, the base of support is widened in
that direction to increase stability
– Staggered stance most stable when catching
while leaning toward the front foot—same
type of stance for tug-of-war, except shift
weight over rear foot
– Boxers, tennis players, baseball batters
Center of Gravity, Stability, and
Human Movement
• Some activities, stability is minimized to
enhance quick movement
– Track sprint start, in the set position, the
sprinter raises COG and moves it forward to
the edge of base of support over hands
– At the starters signal, lifting hands off the track
puts line of action of the force of gravity
outside base of support and the sprinter falls
forward—similar strategy used in swimming
Summary
• Torque or moment of force is the turning
effect created by an eccentric force
• Perpendicular distance between the line of
action of the force and a line parallel that
passes through the axis of rotation is
called the moment arm
• Clockwise torques are negative
• Counterclockwise torques are positive
Summary
• For an object to be balance (i.e. static
equilibrium) all the torques acting on the object
must sum to zero
• COG—Point about which the moments of force
created by the weights of each of the parts of the
object sum to zero
• Stability is affected by the height of COG and its
position relative to edges of the base of support
• Weight also affects stability