Workenergy Relationship

The final approach to studying the kinetics of motion involves laws from a branch of physics dealing with the concepts of work and energy. Since much of the energy in the human body, machines, and on the earth are in the form of heat, these laws are used in thermodynamics to study the flow of heat energy. Biomechanists are interested in how mechanical energies are used to create movement.

Mechanical Energy

In mechanics, energy is the capacity to do work. In the movement of everyday objects, energy can be viewed as the mover of stuff (matter), even though at the atomic level matter and energy are more closely related. Energy is measured in Joules (J) and is a scalar quantity. One Joule of energy equals 0.74 ftlbs. Energy is a scalar because it represents an ability to do work that can be transferred in any direction. Energy can take many forms (for example, heat, chemical, nuclear, motion, or position). There are three mechanical energies that are due to an object's motion or position.

The energies of motion are linear and angular kinetic energy. Linear or transla-tional kinetic energy can be calculated using the following formula: KET = %mv2. There are several important features of this formula. First, note that squaring velocity makes the energy of motion primarily dependent on the velocity of the object. The energy of motion varies with the square of the velocity, so doubling velocity increases the kinetic energy by a factor of 4 (22). Squaring velocity also eliminates the effect of the sign (+ or -) or vector nature of velocity. Angular or rotational kinetic energy can be calculated with a similar formula: KER = %I«2. We will learn more about angular kinetics in chapter 7.

The mathematics of kinetic energy (%mv2) looks surprisingly similar to momentum (mv). However, there are major differences in these two quantities. First, momentum is a vector quantity describing the quantity of motion in a particular direction. Second, kinetic energy is a scalar that describes how much work an object in motion could perform. The variable momentum is used to document the current state of motion, while kinetic energy describes the potential for future interactions. Let's consider a numerical example from American football. Imagine you are a small (80kg) halfback spinning off a tackle with one yard to go for a touchdown. Who would you rather run into just before the goal line: a quickly moving defensive back or a very large lineman not moving as fast? Figure 6.16 illustrates the differences between kinetic energy and momentum in an inelastic collision.

Applying the impulse-momentum relationship is interesting because this will tell us about the state of motion or whether a touchdown will be scored. Notice that both defenders (small and big) have the same amount of momentum (-560 kg-m/s), but because the big defender has greater mass you will not fly backwards as fast as in the collision with the defensive back. The impulse-momentum relationship shows that you do not score either way (negative velocity after impact: V2), but the defensive back collision looks very dramatic because you reverse directions with a faster negative velocity. The work-energy relationship tells us that the total mechani cal energy of the collision will be equal to the work the defender can do on you. Some of this energy is transferred into sound and heat, but most of it will be transferred into deformation of your pads and body! Note that the sum of the energies of the two athletes and the strong dependence of kinetic energy on velocity results in nearly twice (2240 versus 1280 J) as much energy in the collision with the defensive back. In short, the defensive back hurts the most because it is a very high-energy collision, potentially adding injury to the insult of not scoring.

There are two types of mechanical energy that objects have because of their position or shape. One is gravitational potential energy and the other is strain energy. Gravitational potential energy is the energy of the mass of an object by virtue of its position relative to the surface of the earth. Potential energy can be easily calculated with the formula: PE = mgh. Potential energy depends on the mass of the object, the acceleration due to gravity, and the height of the object. Raising an object with a mass of 35 kg a meter above the ground stores 343 J of energy in it (PE = 35 • 9.81 • 1 = 343). If this object were to be released, the potential energy would gradually be converted to kinetic energy as gravity accelerated the object toward the earth. This simple example of transfer of mechanical energies is an example of one of the most important laws of physics: the Law of Conservation of Energy.

The Law of Conservation of Energy states that energy cannot be created or destroyed; it is just transferred from one form to another. The kinetic energy of a tossed ball will be converted to potential energy or possibly strain energy when it collides with another object. A tumbler taking off from a mat has kinetic energy in the vertical direction that is converted into potential energy on the way up, and back into kinetic energy on the way down. A bowler who increases the potential energy of the ball during the

Figure 6.16. Comparison of the kinetic energy (scalar) and momentum (vector) in a football collision. If you were the running back, you would not score a touchdown against either defender, but the work done on your body would be greater in colliding with the smaller defender because of their greater kinetic energy.

Figure 6.16. Comparison of the kinetic energy (scalar) and momentum (vector) in a football collision. If you were the running back, you would not score a touchdown against either defender, but the work done on your body would be greater in colliding with the smaller defender because of their greater kinetic energy.

approach can convert this energy to kinetic energy prior to release (Figure 6.17). In a similar manner, in golf or tennis a forward swing can convert the potential energy from preparatory movement into kinetic energy. A major application area of conservation of energy is the study of heat or thermodynamics.

The First Law of Thermodynamics is the law of conservation of energy. This is the good news: when energy is added into a machine, we get an equal amount of other forms of energy out. Unlike these examples, examination of the next mechanical energy (strain energy) will illustrate the bad news of the Second Law of Thermodynamics: that it is impossible to create a machine that converts all input energy into some useful output energy. In other words, man-made devices will always lose

Figure 6.17. Raising a bowling ball in the approach stores more potential energy in the ball than the kinetic energy from the approach. The potential energy of the ball can be converted to kinetic energy in the downswing.

energy in some non-useful form and never achieve 100% efficiency. This is similar to the energy losses (hysteresis) in strain energy stored in deformed biological tissues studied in chapter 4.

Strain energy is the energy stored in an object when an external force deforms that object. Strain energy can be viewed as a form of potential energy. A pole vaulter stores strain energy in the pole when loading the pole by planting it in the box. Much of the kinetic energy stored in the vaulter's body during the run up is converted into strain energy and back into kinetic energy in the vertical direction. Unfortunately, again, not all the strain energy stored in objects is recovered as useful energy. Often large percentages of energy are converted to other kinds of energy that are not effective in terms of producing movement. Some strain energy stored in many objects is essentially lost because it is converted into sound waves or heat. Some machines employ heat production to do work, but in human movement heat is a byproduct of many energy transformations that must be dissipated. Heat is often even more costly than the mechanical energy in human movement because the cardiovascular system must expend more chemical energy to dissipate the heat created by vigorous movement.

The mechanical properties of an object determine how much of any strain energy is recovered in restitution as useful work. Recall that many biomechanical tissues are viscoelastic and that the variable hysteresis (area between the loading and unloading force-displacement curves) determines the amount of energy lost to unproductive energies like heat. The elasticity of a material is defined as its stiffness. In many sports involving elastic collisions, a simpler variable can be used to get an estimate of the elastic ity or energy losses of an object relative to another object. This variable is called the coefficient of restitution (COR and e are common abbreviations). The coefficient of restitution is a dimensionless number usually ranging from 0 (perfectly plastic collision: mud on your mother's kitchen floor) to near 1 (very elastic pairs of materials). The coefficient of restitution cannot be equal to or greater than 1 because of the second law of thermodynamics. High coefficients of restitution represent elastic collisions with little wasted energy, while lower coefficients of restitution do not recover useful work from the strain energy stored in an object.

The coefficient of restitution can be calculated as the relative velocity of separation divided by the relative velocity of approach of the two objects during a collision (Hatze, 1993). The most common use of the coefficient of restitution is in defining the relative elasticities of balls used in sports. Most sports have strict rules governing the dimensions, size, and specifications, including the ball and playing surfaces. Officials in basketball or tennis drop balls from a standard height and expect the ball to rebound to within a small specified range allowed by the rules. In these uniformly accelerated flight and impact conditions where the ground essentially doesn't move, e can be calculated with this formula: e = (bounce/drop)1/2. If a tennis ball were dropped from a 1-meter height and it rebounded to 58 cm from a concrete surface, the coefficient of restitution would be (58/100)1/2 = 0.76. Dropping the same tennis ball on a short pile carpet might result in a 45-cm rebound, for an e = 0.67. The coefficient of restitution for a sport ball varies depending on the nature of the other object or surface it interacts with (Cross, 2000), the velocity of the collision, and other factors like temperature. Squash players know that it takes a few rallies to warm up the ball and increase its coefficient of restitution.

Also, putting softballs in a refrigerator will take some of the slugging percentage out of a strong hitting team.

Most research on the COR of sport balls has focused on the elasticity of a ball in the vertical direction, although there is a COR in the horizontal direction that affects friction and the change in horizontal ball velocity for oblique impacts (Cross, 2002). The horizontal COR strongly affects the spin created on the ball following impact. This is a complicated phenomenon because balls deform and can slide or rotate on a surface during impact. How spin, in general, affects the bounce of sport balls will be briefly discussed in the section on the spin principle in chapter 8.

Mechanical Work

All along we have been defining mechanical energies as the ability to do mechanical work. Now we must define work and understand that this mechanical variable is not exactly the same as most people's common perception of work as some kind of effort. The mechanical work done on an object is defined as the product of the force and displacement in the direction of the force (W = F • d). Joules are the units of work: one joule of work is equal to one Nm. In the English system, the units of work are usually written as foot-pounds (ft lb) to avoid confusion with the angular kinetic variable torque, whose unit is the lb •ft. A patient performing rowing exercises (Figure 6.18) performs positive work (W = 70 • 0.5 = +35 Nm or Joules) on the weight. In essence, energy flows from the patient to the weights (increasing their potential energy) in the concentric phase of the exercise. In the eccentric phase of the exercise the work is negative, meaning that potential energy is being transferred from the load to the patient's body. Note that the algebraic formula assumes the force applied to the

load is constant over the duration of the movement. Calculus is necessary to calculate the work of the true time-varying forces applied to weights in exercises. This example also assumes that the energy losses in the pulleys are negligible as they change the direction of the force created by the patient.

Note that mechanical work can only be done on an object when it is moved relative to the line of action of the force. A more complete algebraic definition of mechanical work in the horizontal (x) direction that takes into account the component of motion in the direction of the force on an object would be W = (F cos 9) • dx. For example, a person pulling a load horizontally on a dolly given the data in Figure 6.19 would do 435 Nm or Joules of work. Only the horizontal component of the force times the displacement of object determines the work done. Note also that the angle of pull in this example is like the muscle angle of pull analyzed earlier. The smaller the angle of pull, the greater the horizontal component of the force that does work to move the load.

The vertical component of pull does not do any mechanical work, although it may decrease the weight of the dolly or load and, thereby decrease the rolling friction to be overcome. What is the best angle to pull in this situation depends on many factors. Factoring in rolling friction and the strength (force) ability in various pulling postures might indicate that a higher angle of pull that doesn't maximize the horizontal force component may be "biomechanical-ly" effective for this person. The inertia of the load, the friction under the person's feet, and the biomechanical factors of pulling from different postures all interact to determine the optimal angle for pulling an object. In fact, in some closed kinematic chain movements (like cycling) the optimal direction of force application does not always maximize the effectiveness or the component of force in the direction of motion (Doorenbosch et al., 1997).

Mechanical work does not directly correspond to people's sense of muscular effort. Isometric actions, while taking considerable effort, do not perform mechanical

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