chapter 5 torques and moments of force maintaining equilibrium or changing angular motion

Chapter 5

Torques and Moments of ForceMaintaining 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 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


• Torque is defined mathematically as:

• T = Fr

• T = torque (or moment of force)

• F = Force (Newtons)

• r = moment arm (meters)


• 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 = 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?


• 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


• 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?


• 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


• 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


• 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)


• 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)


• 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


• 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?


• 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


• 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


• 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


• 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


• 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


• 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


• 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


• Sum of the moments about the axis equals zero

• ΣTa= 0

• 0 = (P x h) – (W x b)

• P x h = W x b


• 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


• 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


• 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


• 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?


• 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?


• 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


• 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


• 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)


• 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


• 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


• 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

chapter 5 torques and moments of force maintaining equilibrium or changing angular motion

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