## W 80 cos 25 6 435 Nm

work. This dependence on the object's displacement of mechanical work makes the work-energy relationship useful in biome-chanical studies where the motion of an object may be of more interest than temporal factors.

This brings us to the Work-Energy Relationship, which states that the mechanical work done on an object is equal to the change in mechanical energy of that object. Biomechanical studies have used the work-energy relationship to study the kinetics of movements. One approach calculates the changes in mechanical energies of the segments to calculate work, while the other calculates mechanical power and integrates these data with respect to time to calculate work. The next section will discuss the concept of mechanical power.

### Mechanical Power

Mechanical power is an important kinetic variable for analyzing many human movements because it incorporates time. Power is defined as the rate of doing work, so mechanical power is the time derivative of mechanical work or work divided by time (P = W/t). Note that a capital "P" is used because lower-case "p" is the symbol for mo mentum. Typical units of power are Watts (one J/s) and horsepower. One horsepower is equal to 746 W. Maximal mechanical power is achieved by the right combination of force and velocity that maximizes the mechanical work done on an object. This is clear from the other formula for calculating power: P = F • v. Prove to yourself that the two equations for power are the same by substituting the formula for work W and do some rearranging that will allow you to substitute v for its mechanical definition.

If the concentric lift illustrated in Figure 6.18 was performed within 1.5 seconds, we could calculate the average power flow to the weights. The positive work done on the weights was equal to 35 J, so P = W/t = 35/1.5 = 23.3 W. Recall that these algebraic definitions of work and power calculate a mean value over a time interval for constant forces. The peak instantaneous power flow to the weight in Figure 6.18 would be higher than the average power calculated over the whole concentric phase of the lift. The Force-Motion Principle would say that the patient increased the vertical force on the resistance to more than the weight of the stack to positively accelerate it and would reduce this force to below the weight of the stack to gradually stop the weight at the end of the concentric phase. Instantaneous power flow to the weights also follows a complex pattern based on a combination of the force applied and the motion of the object.

What movements do you think require greater peak mechanical power delivered to a barbell: the lifts in the sport of Olympic weight lifting or power lifting? Don't let the names fool you. Since power is the rate of doing work, the movements with the greatest mechanical power must have high forces and high movement speeds. Olympic lifting has mechanical power outputs much higher than power lifting, and power lifting is clearly a misnomer given the true definition of power. The dead lift, squat, and bench press in power lifting are high-strength movements with large loads but very slow velocities. The faster movements of Olympic lifting, along with smaller weights, clearly create a greater power flow to the bar than power lifting. Peak power flows to the bar in power lifting are between 370 and 900 W (0.5-1.2 hp), while the peak power flow to a bar in Olympic lifts is often as great as 4000 W or 5.4 hp (see Garhammer, 1989). Olympic lifts are often used to train for "explosive" movements, and Olympic weight lifters can create significantly more whole-body mechanical power than other athletes (McBride, Triplett-McBride, Davie, & Newton, 1999).

Many people have been interested in the peak mechanical power output of whole-body and multi-segment movements. It is believed that higher power output is critical for quick, primarily anaerobic movements. In the coaching and kinesiolo-gy literature these movements have been described as "explosive." This terminology may communicate the point of high rates of force development and high levels of both speed and force, but a literal interpretation of this jargon is not too appealing! Remember that the mechanical power output calculated for a human movement will strongly depend on the model (point mass, linked segment, etc.) used and the time interval used in the calculation. The average or instantaneous power flows within the body and from the body to external objects are quite different. In addition, other bio-mechanical factors affect how much mechanical power is developed during movements.

The development of maximal power output in human movement depends on the direction of the movement, the number of segments used, and the inertia of the object. If we're talking about a simple movement with a large resistance, the right mix of force and velocity may be close to 30 to 45% of maximal isometric strength because of the Force-Velocity Relationship (Izquierdo et al., 1999; Kaneko et al.f 1983; Wilson et al., 1993). Figure 6.20 shows the in vitro concentric power output of skeletal muscle derived from the product of force and velocity in the Force-Velocity Relationship. In movements requiring multi-joint movements, specialized dynamometer measurements indicate that the best resistances for peak power production and training are likely to be higher than the 30-45% and differ between the upper and lower extremi-

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