Topic 06 – Torques and Moments of Force

Topic 06 – Torques and Moments of Force

Lesson Learning Outcomes

• Define torque (moment of force)
• Define static equilibrium and list the equations of static equilibrium
• Determine the resultants of 2 or more torques
• Determine if an object is in static equilibrium, when the forces and torques acting on the object are known
• Determine unknown force (or torque) acting on an object is in static equilibrium and all other forces and torques acting on the object are known
• Define center of gravity and estimate the location of COG of an object/body.

Guiding Questions

• Why is it important to understand torques and how can it be applied to human movements in sport and exercises?
• How is it possible to determine an unknown torque on an object, if the other forces acting the object is known, and object is in static equilibrium?
• Why is it important to understand Centre of Gravity and how can it be applied to improve human movements in sport and exercises?

WHAT ARE TORQUES?

The turning effect produced by a force is called a torque. The torque produced by a force may also be called a moment of force. This term may also be simplified to “moment”.

Example: Moving a book

In (a), the resultant force acting on the book was directed through the center of gravity of the book. An external force directed through the center of gravity of an object is called a centric force. The effect of a centric force is to cause a change in the linear motion of the object, as predicted by Newton’s second law of motion.

In (b) the resultant force acting on the book was not directed through the center of gravity of the book. An external force not directed through the center of gravity is called an eccentric force. The effect of an eccentric force is to cause a change in the linear and angular motions of an object. The torque produced by the eccentric force caused the book to rotate.

In (c), a pair of forces acted on the book. These forces were equal in size but opposite in direction and noncolinear. A pair of such forces is called a force couple. The effect of a force couple is to cause a change only in the angular motion of an object. The resultant of the 2 forces in a force couple is the force of zero, therefore no change in linear motion occurs. The torque produced by the force couple caused the book to rotate.

In general:

1. a centric force will cause or tend to cause a change in the linear motion of an object;
2. an eccentric force will cause or tend to cause a change in linear and angular motions of an object
3. a force couple will cause or tend to cause only a change in the angular motion of an object.

• Mathematical Definition of Torque

The torque produced by a force is directly proportional to the size of the force as well as the distance between the line of action of the force and the point about which the object tends to rotate (the axis of rotation).

A torque is the turning effect produced by a force and is equal to the product of the magnitude of the force and the axis of rotation of the object (or the axis about which the object will tend to rotate). This distance between the line of action of the force and the axis of rotation is the perpendicular distance between the line of action of the force and a line parallel to it that passes through the axis of rotation. This distance is sometimes referred to as the moment arm of the force.

The SI unit of measurement for torque is units of force x units of length (N x m = Nm).

Torque is a vector quantity, because the turning effect is around a specific axis that is directed in a specific direction. The conventional approach is to indicate counterclockwise torques as positive (+) and clockwise torques as negative (-).

• Examples of How Torques Are Used

Doorknobs or door handles are generally located on the opposite side of the door from the hinges. This location makes the moment arm, r in the torque equation, large so that the force required to create a large enough torque to swing open the door is small.

A given size torque can be created with a large force and a small moment arm or with a small force and a large moment arm. Tools below increases the torque that human are able to produce by using a relative larger moment arm. All of these tools have handles that increase the length of the moment arm of the force, thus increasing the torque applied to the screw, nut, and so on.

In any sport in which we turn, spin or swing, torque must be created to initiate these turns or swings. The holds used in wrestling are great examples of torque that are used to turn an opponent. You try to turn over your opponent by pushing down on his head with your hand and use your arm to lift up under his shoulder, creating a force couple. To counter this, your opponent can create a torque with his other arm by abducting it, so that it is perpendicular to his body.

• Muscular Torque

Muscle creates the torques that turn our limbs. A muscle creates a force that pulls on its points of attachment to the skeletal system when it contracts. The line of action (or the line of pull) of a muscle force is along a line joining its attachments and is usually indicated by the direction of its tendons. The bones that a muscle attaches to are within the limbs on either side of a joint, or 2 or more joints in some cases. When a muscle contracts, it creates a pulling force on these limbs. Because of the line of action of the muscle force is some distance from the joint axis, a moment arm exists and torques about the joint axis are produced by the muscle force on the limbs where the muscle attaches. Changing the angle at the joint changes the moment arm of the muscles that cross that joint.

• Strength Training Devices and Torque

When you do an arm curl exercise, as you lift the dumbbell, its weight produces a torque around the elbow joint as well, but this torque tends to rotate the forearm in the opposite direction. It creates a torque that would tend to extend your forearm at the elbow joint. The dumbbell doesn’t get heavier, but the torque gets larger as the elbow is flexed. This occurs because the line of action of the dumbbell’s weight moves farther from the elbow joint as the exercise is performed, thus increasing the moment arm. The moment arm of the dumbbell is at its maximum and the torque is greatest when the forearm is horizontal, and elbow is at 90deg.

Example of weightlifting machine – Leg Extension Machine

The resistance force is provided by a stack of weights, and the force is then transmitted to the arm of the machine via a cable attached to the middle of the arm. As the exercise is performed, the cable pulls on the machine arm. A torque is created about the axis of the machine arm because the cable pulls some distance away from this axis. As the exercise is performed, the line of action of the cable force changes, and the moment arm gets smaller as the leg reaches full extension at the knee. This resistance torque is thus largest at the starting position with the knee at 90deg, and gets smaller as the knee extends.

The design goal of some machines is to provide a constant resistance torque throughout the range of motion of exercise. Other machines are designed to provide a resistance torque that varies in proportion to the changes in the moment arm of the muscle being exercised as the exercise is performed.

FORCES AND TORQUES IN EQUILIBRIUM

If an object is at rest, it is described as in a state of static equilibrium. For an object to be in static equilibrium, the external forces acting on it must sum to zero. A net force of zero is not the only condition of static equilibrium; A net force of zero ensures that no change will occur in the linear motion of an object, but it does not constrain the object’s angular motion. The net torque acting on an object must be zero to ensure that no changes occur in the angular motion of the object.

For an object to be in static equilibrium, the external forces must sum to zero and the external torque (about any axis) must sum to zero as well.

• Net Torque

Torques that acted around the same axis could be added or subtracted algebraically, just like forces that act in the same direction. In a planar situation, then, we compute a net torque by summing the torques that act on an object.

Example: Ruler, Rubber Eraser and Coins

Balance the ruler on the edge of the eraser. If you don’t have an eraser, find something with a flat surface about 1/4 in. (0.6 cm) wide that you can balance the ruler on. If the ruler is 12 in. (30 cm) long it probably balances with the eraser at 6 in. (15 cm). Now place a penny on the ruler 5 in. (13 cm) to the left of the eraser. Does the ruler stay balanced (is it in a state of static equilibrium)? No. Why not?

The penny in the example above created a counter clockwise torque about the eraser that caused the ruler to rotate counterclockwise. The moment arm of the penny was 5 in. The force created by the penny was the weight of the penny. The torque created about the eraser was one pennyweight (p) x 5 in. In this case, the net torque acting on the ruler about an axis through the eraser was caused only by the weight of the penny.

This torque is positive because it tended to cause a rotation of the ruler in the counterclockwise direction. When you place 2 more pennies as the diagram below on the right of the ruler, the ruler balances on the eraser, and is in static equilibrium.

The ruler balances, because the net torque is zero:

Clear the pennies off the ruler and stack 4 pennies on the ruler 3 in. (about 8cm to thr right of the eraser. If we have 2 pennies left to use to balance thr ruler, where would you stack them?

Two pennies will create a counterclockwise torque of 12 p·in. (to counter the 12 p·in. of clockwise torque created by the four pennies) if you stack them 6 in. to the left of the eraser (the negative sign before the 6 in. indicates that the moment arm is to the left of the axis). Two pennies times 6 in. equals 12 p·in. of torque in the counterclockwise direction.

• Muscle Force Estimates Using Equilibrium Equations

Conditions for static equilibrium can allow us to estimate the forces our muscles produce to lift of hold up objects.

Example: 9kg dumbbell

Suppose you are holding a 20 lb (9 kg) dumbbell in your hand and your elbow is flexed 90° so that your forearm is parallel to the floor. If the moment arm of this dumbbell is 12 in. (30 cm) about the elbow  joint axis, what torque is created by this dumbbell about your elbow joint axis? The torque is:

To hold up this dumbbell, your elbow flexor muscles must create a counterclockwise torque of 240lb.in. If the moment arm of these muscles is 1 in., what force must they pull with to hold the dumbbell in position?

Your elbow flexor muscles must create a force of 240 lb (about 1068 N) just to hold up a 20 lb dumbbell! This seems to be too large a force, but our muscles are arranged so that their moment arms about the joints are short. This means that they must create relatively large forces to produce practically effective torques about the joints.

• More Examples of Net Torque

What net torque acts on this vaulter around an axis through his center of gravity? Is the vaulter in equilibrium? The 500 N force acting at the vaulter’s left hand has a moment arm of 0.50 m about his center of gravity. This force creates a clockwise torque about the vaulter’s center of gravity. The 1500 N force acting at the vaulter’s

right hand has a moment arm of 1.00 m about his center of gravity. The torque created by this force about the vaulter’s center of gravity is also clockwise. The vaulter’s weight of 700 N acts through his center of gravity, and thus the moment arm of this weight is zero, so it does not create any torque about the vaulter’s center of gravity.

Mathematically, the net torque about a transverse axis through the vaulter’s center of gravity is:

The negative sign on this net torque indicates that it acts in a clockwise direction. The 1750Nm net torque produces a turning effect that tends to rotate the vaulter clockwise onto his back, as if he were doing a backward somersault.

In Figure 5.11, 300 N force acting on the vaulter’s left hand has a moment arm of 0.50 m and still creates a clockwise torque about the vaulter’s center of gravity. The 500 N force acting on the vaulter’s right hand also has a moment arm of 0.50 m, but now it creates a counterclockwise torque about the vaulter’s center of gravity. The net torque is therefore:

The positive sign on this net torque indicates that it acts in the counterclockwise direction. The turning effect produced by the forces acting on the vaulter thus tends

to rotate the vaulter counterclockwise, as if he were doing a forward somersault or to slow down the vaulter’s clockwise rotation. The latter is likely the case, since the

pole vaulter was rotating clockwise earlier in the vault. This counterclockwise torque will stop the clockwise rotation as the vaulter lines up his or her body with the

pole in preparation for going over the crossbar. It will eventually cause the vaulter to rotate counterclockwise. This counterclockwise rotation is helpful as the vaulter’s body rotates over the crossbar during the clearance.

CENTER OF GRAVITY

Center of gravity is the 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 is the point in a body or system of bodies at which, for certain purposes, the entire mass may be assumed to be concentrated. For bodies near the earth, this coincides with the center of gravity.

The center of gravity is a useful concept for analysis of human movement because it is the point at which the entire mass or weight of the body may be considered to be concentrated. So the force of gravity acts downward through this point.

If a net external force acts on a body, the acceleration caused by this net force is the acceleration of the center of gravity. If no external forces act on an object, the center of gravity does not accelerate. When we are interpreting and applying Newton’s laws of motion, it is the center of gravity of a body whose motions are ruled by these laws.

Locating the Center of gravity

If some of the elemental parts of an object move or change position, the center of gravity of the object moves as well.

If an elemental part of an object is removed from the object, the center of gravity of the object moves away from the point of removal.

If mass is added to an object, the center of gravity of the object moves toward the location of the added mass.

Mathematical Determination of the Center-of-Gravity Location

If the weights and locations of the elemental parts that make up an object are known, the center-of-gravity location can be computed mathematically. The definition of center of mass indicated that it is the point at which the entire mass (or weight) may be considered to be concentrated.

By this definition, a ruler with six pennies distributed on it at 2 in. (5 cm) intervals is equivalent to a ruler with six pennies stacked on it at one location if that location is the center of gravity of the first ruler. The sum of the moments of force created by each of the elemental weights, the pennies in this case, equals the moment of force created by the total weight stacked at the center-of-gravity location. Mathematically, this can be expressed as

Let’s use the end of the first ruler shown in figure 5.12 as the axis about which the moments of force will be measured.

To determine the location of the center of gravity mathematically, we use the relationship between the sum of the moments of force created by the elemental weights and the moment of force created by the sum of the elemental weights (i.e., they are equal). Stated more simply, the sum of the moments equals the moment of the sum.

Place three pennies on the ruler at the 1 in. (2.5 cm) mark and seven pennies on the ruler at the 8 in. (20 cm) mark. Mathematically, we can solve for the location of the center of gravity as we did in the previous example.

With the pennies and ruler, we found the center of gravity along one dimension only. For more complex objects, the center-of-gravity location is defined by three dimensions, because most objects occupy space in three dimensions. To determine the center-of-gravity location for an object in three dimensions, the procedure we used in the previous two examples is repeated for each dimension, with gravity assumed to be acting in a direction perpendicular to that dimension.

Center of Gravity of the Human Body

Because your left and right sides are symmetrical, your center of gravity lies within the plane that divides your body into left and right halves (the midsagittal plane). If you lift your left arm away from your side, your center of gravity shifts to the left.

Although your body is not symmetrical from front to back, the center of gravity lies within a plane that divides your body into front and back halves (the frontal plane). This plane passes approximately through your shoulder and hip joints and slightly in front of your ankle joints. If you raise your arms in front of you, your center of gravity will be moved forward slightly.

The location of the center of gravity in the vertical dimension is more difficult to estimate. The center of gravity from top to bottom lies within a plane that passes horizontally through your body 1 to 2 in. (2.5 to 5 cm) below your navel, or about 6 in. (15 cm) above your crotch. This plane is slightly higher than half of your standing height, about 55% to 57% of your height. If you reach overhead with both arms, your center of gravity will move upward slightly (about 2 or 3 in. [5 to 8 cm]). Someone with long legs and muscular arms and chest has a higher center of gravity than someone with shorter, stockier legs. A woman’s center of gravity is slightly lower than a man’s because women have larger pelvic girdles and narrower shoulders relative to men. A woman’s center of gravity is approximately 55% of her height from the ground, whereas a man’s center of gravity is approximately 57% of his height from the ground. Infants and young children have higher centers of gravity relative to their height because of their relatively larger heads and shorter legs.

The center of gravity of the human body will move from the position just described if the body parts change their positions. For each limb movement, the center of gravity of the entire body shifts slightly in the direction of that movement. How much it moves depends on the weight of the limb that moved and the distance it moved. In the below examples, the center of gravity may lie outside of your body.

Center of Gravity and Performance

Consider a jump and reach test with three scenarios:

1. From a standing position, jump up and reach up to the wall as high as you can with one arm stretched up and other one down at your side
2. From a standing position, jump and reach with both arms stretched overhead
3. From a standing position, jump and reach with both arms stretched overhead and lift your knees and leg so your heels touch your buttocks.

Result:

1. Highest
2. 1 or 2 inch lower
3. 4 -6 inch lower

When you jumped up in the air during the jump-and-reach tests, the only external force acting on you was the force of gravity (your weight). This force causes your center of gravity to accelerate downward at a constant rate.  You were a projectile once your feet left the ground. The motion of your limbs when you were in the air could not alter the motion of your center of gravity because they were not pushing or pulling against anything external to your body. The net force acting on you was still only the downward pull of gravity. The path that your center of gravity followed was not changed by your limb actions, but these actions caused the motion of your other limbs and trunk to change. Lifting your arms caused something else to move downward in order for your center of gravity to continue moving along its parabolic path. When you raised your arms and legs, your head and trunk got lower to compensate for this movement.

First, let’s assume that the center of gravity reached the same peak height. If this is the case, then to maximize the reach of one hand (its height above the ground), you want to maximize the vertical distance between the center of gravity and the outstretched arm. Raising both arms, or both arms and both legs, moves the center of gravity closer to the head and thus closer to the reaching hand. Keeping all of the limbs and body parts (with the exception of the reaching arm and hand) as low as possible relative to the center of gravity maximizes the distance from the reaching hand to the center of gravity.

Let’s consider one final jumping activity. How do basketball players, dancers, figure skaters, and gymnasts appear to “hang in the air”? If we follow the path of the jumper’s center of gravity, it does indeed rise and fall through a parabolic curve. But the jumper’s head and trunk appear to be suspended at the 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 center of gravity, so the head and trunk do not rise appreciably.

Center of Gravity and Stability

Stability is the capacity of an object to return to equilibrium or to its original position after it has been displaced.

Factors affecting Stability

The stability of an object is affected by the height of the center of gravity, the size of the base of support, and the weight of the object.

Figure 5.17 shows examples of various stances and their bases of support. Which of these stances are most stable? Which of these stances are least stable?

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. The external forces acting on the book include the book’s weight, W, acting through its center of gravity, cg; the toppling force, P; and the reaction force, R. If we examine the situation in which the toppling force is just large enough that the book is almost starting to move, the free-body diagram shown in figure 5.18 is appropriate. If moments of force are measured about an axis, a, through the lower left corner of the book, the sum of moments about this point is zero

To increase stability, minimize variables on the left side of the equation and maximize variables on the right side of the equation.

The dimension, b, is related to the size of the base of support, but it may be smaller or larger depending on which direction the toppling force comes from. Figure 5.19 demonstrates this for objects of various shapes.

The triangular block is less stable and more likely to be tipped over if the toppling force is directed to the left (figure 5.19a) rather than to the right (figure 5.19b).

Stability is directional. An object can be more stable in one direction than in 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 in which the toppling force is pushing or pulling.

On the left side of equation 5.4, the toppling force, P, is a factor that is not related to any characteristic of the object. The moment arm of this toppling force, h, is a characteristic of the object. This distance is related to the height of the object, which is also related to the height of the center of gravity. So a lower center of gravity, which implies a lower height and a shorter moment arm for the toppling force, increases stability.

Stability and Potential Energy

A better way of explaining why center-of-gravity height affects stability uses the concepts of work and potential energy.

Consider the block shown in figure 5.20. As long as the center of gravity 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. But when the block is tipped past the configuration shown in figure 5.20b, where the center of gravity lies directly over the supporting corner, the moment of force created by the weight changes direction and becomes a toppling moment that causes the block to topple, as shown in figure 5.20c.

To move the block from its stable position (figure 5.20a) to the brink of instability (figure 5.20b), the center of gravity had to be raised a distance, 𝚫h. Work was required to do this, and the potential energy of the block increased.

Figure 5.21 shows three blocks of the same shape and weight but with differing center-of-gravity heights. The higher the center of gravity, the smaller this vertical displacement, thus the smaller the change in potential energy and the smaller the amount of work done. So a block with a lower center of gravity is more stable because more work is required to topple it.

If the distance from the line of gravity to the edge of the base of support about which toppling will occur (the moment arm of the weight) is increased, the vertical displacement that the center of gravity goes through before the object topples also increases, so the object is more stable. Figure 5.22 shows two blocks with identical center-of-gravity heights but different horizontal distances from the line of gravity to the edge of the base of support.

The most stable stance or position that an object or person can be in is the one that minimizes potential energy.

Positions that place the center of gravity below the points of support are more stable than positions that place the center of gravity above the base of support.

Center of Gravity, Stability and Human Movement

The human body is not rigid, and its center-of-gravity position and base of support can change with limb movements. Humans can thus control their stability by changing their stance and body position.

During the first period of a wrestling match, the two wrestlers are standing, and each is trying to take the other down.

To maximize stability (while still maintaining the ability to move), the wrestler crouches to lower his center of gravity and widens his base of support by placing his feet slightly wider than shoulder-width apart in a square stance (figure 5.24a) or by placing one foot in front of the other, again slightly farther than shoulder-width apart, in a staggered stance (figure 5.24b).

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 center of gravity as much as possible (figure 5.25).

If a heavy medicine ball were thrown to you, the most stable stance for you to be in to catch the ball would be a staggered stance, with one foot in front of the other in line with the direction of the throw and your body leaning toward the front foot (figure 5.26). This is a popular stance in many sports, not only because it allows momentum to be reduced or increased by force application over a long time, but also because it is a more stable position.

The size of our base of support is limited by our shoe size and the stances being adopted.

In sport, skis increase stability forward and backward. In physical rehabilitation and medicine, crutches, canes, walkers, and so on are used to increase the base of support and stability of people who are injured, sick, or infirm.

In some activities, stability is minimized to enhance quick movement. For instance, in a track sprint start, in the set position, the sprinter raises her center of gravity and moves it forward to the edge of the base of support over her hands. At the starter’s signal, lifting her hands off the track puts her line of gravity well in front of her base of support, and the sprinter falls forward. A similar strategy is used in swimming starts.

SUMMARY:

• The turning effect created by a force is a moment of force, also called a torque.
• A torque is equal to the magnitude of the force times the perpendicular distance between the line of action of the force and the axis of rotation about which the torque is being measured. This perpendicular distance is called the moment arm of the force.
• Torque is a vector quantity. Its direction is defined by the orientation of the axis of rotation and the sense of the torque (CW or ACW) about the axis of rotation.
• The SI Units for torque are Newton x meters (Nm)
• For an object to be in state of equilibrium, external forces acting on it must sum to zero and moments of those forces must also sum to zero.
• To locate an object center or gravity, you can balance, suspend it or spin it.
• In the human body, the center of gravity lies on the midline left to right and front to back and between 55% – 57% of a person’s height above the ground when the person is standing in anatomical position.
• Stability increases as the center of gravity is lowered and moved farther from any edge of the base of support. Increasing weight also increase stability.