## B

Figure 7.9. The distribution of mass most strongly affects moment of inertia, so wheel A with mass close to the axle would have much less resistance to rotation than wheel B. Wheel A would make it easier for a cyclist to make quick adjustments of the wheel back and forth to balance a unicycle.

batting style affects optimal bat mass (Bahill & Freitas, 1995) and moment of inertia (Watts & Bahill, 2000) for a particular batter.

You can now see that the principle of inertia can be extended to angular motion of biomechanical systems. This application of the concepts related to moment of inertia are a bit more complex than mass in linear kinetics. For example, a person putting on snowshoes will experience a dramatic increase (larger than the small mass of the shoes implies) in the moment of inertia of the leg about the hip in the sagittal plane because of the long radius for this extra mass. A tennis player adding lead tape to the head of their racket will more quickly modify the angular inertia of the racket than its linear inertia. Angular inertia is most strongly related to the distribution of mass, so an effective strategy to decrease this inertia is to bring segment masses close to the axis of rotation. Coaches can get players to "compact" their extremities or body to make it easier to initiate rotation.

NEWTON'S ANGULAR ANALOGUES

Newton's laws of motion also apply to angular motion, so each may be rephrased using angular variables. The angular analogue of Newton's third law says that for every torque there is an equal and opposite torque. The angular acceleration of an object is proportional to the resultant torque, is in the same direction, and is inversely proportional to the moment of inertia. This is the angular expression of Newton's second law. Likewise, Newton's first law demonstrates that objects tend to stay in their state of angular motion unless acted upon by an unbalanced torque. Biomecha-nists often use rigid body models of the human body and apply Newton's laws to calculate the net forces and torques acting on body segments.

This working backward from video measurements of acceleration (second derivatives) using both the linear and angular versions of Newton's second law is called inverse dynamics. Such analyses to understand the resultant forces and torques that create movement were first done using laborious hand calculations and graphing (Bressler & Frankel, 1950; Elftman, 1939), but they are now done with the assistance of powerful computers and mathematical computation programs. The resultant or net joint torques calculated by inverse dynamics do not account for co-contraction of muscle groups and represent the sum of many muscles, ligaments, joint contact, and other anatomical forces (Winter, 1990).

Despite the imperfect nature of these net torques (see Hatze, 2000; Winter 1990), inverse dynamics provides good estimates of the net motor control signals to create human movement (Winter & Eng, 1995), and can detect changes with fatigue (Apriantono et al., 2006) or practice/learning (Schneider et al., 1989; Yoshida, Cauraugh, & Chow, 2004). Figure 7.10 illus-

Figure 7.10. The net joint hip (thick line) and knee (thin line) joint torques in a soccer kick calculated from inverse dynamics. The backswing (BS), range of deepest knee flexion (DKF), forward swing (FS), and impact (IMP) are illustrated. Adapted with permission from Zernicke and Roberts (1976).

trates the net joint torques at the hip and knee in a soccer toe kick. These torques are similar to the torques recently reported in a three-dimensional study of soccer kicks (Nunome et al., 2002). The kick is initiated by a large hip flexor torque that rapidly decreases before impact with the soccer ball. The knee extensor torque follows the hip flexor torque and also decreases to near zero at impact. This near-zero knee extensor torque could be expected because the foot would be near peak speed at impact, with the body protecting the knee from hyperextension. If the movement were a punt, there would usually be another rise and peak in hip flexor torque following the decline in knee torque (Putnam, 1983). It is pretty clear from this planar (2D) example of inverse dynamics that the hip flexor musculature may make a larger contribution to kicking than the knee extensors. It is not as easy to calculate or interpret 3D kinetics since a large joint torque might have a very small resistance arm and not make a large contribution to a desired motion, or a torque might be critical to positioning a segment for another torque to be able to accelerate the segment (Sprigings et al., 1994; Bahamonde, 2000).

The resultant joint torques calculated in inverse dynamics are often multiplied by the joint angular velocity to derive net joint powers. When the product of a net joint torque and joint angular velocity are positive (in the same direction), the muscle action is hypothesized to be primarily concentric and generating positive work. Negative joint powers are hypothesized to represent eccentric actions of muscle groups slowing down an adjacent segment. These joint powers can be integrated with respect to time to calculate the net work done at joints. Other studies first calculated mechanical energies (kinetic and potential energies), and summed them to estimate work and eventually calculate power. Unfortunately, these summing of mechani cal energy analyses do not agree well with direct calculation of joint power from torques because of difficulties in modeling the transfer of mechanical energies between external forces and body segments (Aleshinsky, 1986a,b; Wells, 1988) and coac-tivation of muscles (Neptune & van den Bogert, 1998).

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