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Chapter 5

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

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Chapter 5

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  1. Chapter 5 Torques and Moments of Force Maintaining Equilibrium or Changing Angular Motion

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. Mathematical Definition of Torque • Torque is defined mathematically as: • T = Fr • T = torque (or moment of force) • F = Force (Newtons) • r = moment arm (meters)

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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?

  13. 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

  14. 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

  15. 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

  16. 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)

  17. 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?

  18. 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?

  19. 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?

  20. 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

  21. 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?

  22. 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

  23. 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)

  24. 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

  25. 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

  26. 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

  27. 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

  28. 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)

  29. 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)

  30. 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

  31. 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?

  32. 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

  33. 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

  34. 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

  35. 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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