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Chapter 5

- Torques and Moments of Force
- Maintaining Equilibrium or Changing Angular Motion

Objectives

- 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 forceCauses 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

coupleCauses a change only in the angular motion

of an object - Resultant of the two forces in a force couple is

zero

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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 armPerpendicular distance between the

line of action of the force and a line parallel

to it that passes through the axis of rotation

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Mathematical Definition of Torque

- 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 quantityTurning 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

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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

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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

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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?

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Muscular Torque

- Changing the angle at the joint changes the

moment arm of the muscles that cross that

jointPartially 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 doesnt 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

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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?

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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 F x r
- 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 pole-vaulter?
- 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?

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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

gravitycreates clockwise torque - 1500 N force acting on the vaulters right hand

has a moment arm of 1.0 m about his/her center of

gravityalso clockwise - Vaulters weight of 700 N acts through center of

gravitymoment arm is zero so zero torque

More Examples of Net Torque

- ST S(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?

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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

gravitystill creates clockwise torque - 500 N force acting on the vaulters right hand has

a moment arm of .5 m about his/her center of

gravitybut now counterclockwise

More Examples of Net Torque

- ST S(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 concentratedfor 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

partsin human body represented by limbs, trunk,

and head - Force of gravity pulls downward on each of these

smaller elemental partssum 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

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Mathematical Determination of the COG Location

- If you closed your eyes and picked up both rulers

by the end, they would feel identicalboth 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
- ST S(W x r) (SW) x rcg
- W weight of one element
- r moment arm of an individual element
- SW 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?

(No Transcript)

Mathematical Determination of the COG Location

- Sum the torques of the weights about the right

end of the barbell - ST 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 - ST 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 navel55-57 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

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Center of Gravity of the Human Body

- Womans COG slightly lower than mans because

women have larger pelvic girdles/narrower

shoulders - Infants and children have higher COGs 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

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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

(No Transcript)

Center of Gravity of the Human Body

- Basketball player vs. volleyball player
- What about hang time?
- COG follows parabolic path, but jumpers head and

trunk appear to be suspended at same height

during the middle stage of the leap - During this time, the jumpers legs and arms rise

and then fallthese movements account for the

rise and fall of the COG, so the head and trunk

do not rise appreciably

(No Transcript)

COG and Stability

- Stabilitythe 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 supportarea within the lines connecting

the outer perimeter of each of the points of

support - Weight

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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
- Books weight, W, acting through its COG
- Toppling force, P
- Friction force, Ff
- Reaction Force, R
- Axis through the lower left corner of the book

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Factors Affecting Stability

- Sum of the moments about the axis equals zero
- STa 0
- 0 (P x h) (W x b)
- P x h W x b

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 COGlower 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 objects weight

will increase stabilityrelated to the size of

the base of supportdirection of toppling force

important

Factors Affecting Stability

- Stability is directionalsee 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

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Stability and Potential Energy

- Concepts of work and potential energy explain why

COG height affects stability - Figure 5.20As 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

(No Transcript)

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, ?hWork was required to do this, and

the potential energy of the block increased - What if the COG is higher or lower?

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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?

(No Transcript)

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 stableGymnast 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

(No Transcript)

Center of Gravity, Stability, and Human Movement

- Human body not rigidCOG position and base of

support can change with limb movements - Humans can control stability by changing stance

and body position - Example
- WalkingLean forward until your line of gravity

falls in front of your feet and you lose your

stabilityYou begin to fall forward, and you step

with one foot to catch your fall and reestablish

your stabilityWalking 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

(No Transcript)

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 footsame type of stance

for tug-of-war, except shift weight over rear

foot - Boxers, tennis players, baseball batters

(No Transcript)

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

forwardsimilar 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 - COGPoint 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

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