Applications to Human Body Dynamics

The most fundamental step in the analysis of movement and motion is drawing a free-body diagram showing all external forces acting on an object. If the free body under consideration is part of a body segment, then external forces to be shown include those forces that the rest of the body apply on the body segment under consideration. Although such forces are treated as internal forces in the overall motion of the entire body, they need to be considered as external forces when the movement of an individual body segment is studied.

Some of the external forces acting on an object (part of an object) may be known in both in direction and magnitude. An example of such a force is the gravitational force. In other cases, we might know something about the direction of an external force. When the friction between objects can be neglected, direction of contact force becomes perpendicular to the surface of contact. Frictional force is always in the direction that opposes relative sliding on the surface of contact. There are also propul sive forces that actuate movement and motion. As in the case of bicycling, the propulsive force is in the direction of motion. In speed skating, the propulsive force lies in a plane that is at right angles to the gliding direction. In the following, we present examples concerning the planar motion of humans.

Example 4.7. Diving. Competitive dives in swimming involve several turns before the diver enters the water with as little splash as possible. In one such dive, the diver began the dive with hands at his sides and at the end of the board with his back toward the water. He quickly adopted the layout position where his arms extended in the line of the body, and then assumed the tuck position in which the thighs and the lower legs are pulled in toward the trunk (Fig. 4.10). The angular velocity of the athlete

Figure 4.10a-d. Planar diving in which the diver changes shape from fully extended (a) to tucked configuration (d) through intermediate shapes (b) and (c).

in the layout position was w = ā2^ (s_1). Determine his angular velocity in the tuck position. The moment of inertia Ic of the athlete in the layout position was 134 lb-in-s2 and in the tuck position 34 lb-in-s2.

Solution: Once the swimmer is airborne the only force acting on him is the gravitational force passing through his center of mass, and as there is no external moment acting on him while he is airborne, his angular momentum with respect to his center of mass must be constant:

= 134 lb-in-s2 X ā2^ (sāx) e3 = 34 lb-in-s2 X w e3 w = ā24.75 rad/s

According to this finding, the rate of rotation of the athlete increases approximately fourfold when he assumes the tuck position. In diving, the angular momentum is determined at the time of the takeoff. Once the angular momentum is set, then it remains constant until the diver encounters the water. Note that during the fraction of a second when the swimmer switches from the layout position to the tucked position, his body is changing shape and therefore cannot be idealized as a rigid body. In that brief time period, Hc = Ic w e3 will not hold, because it was derived under the assumption that all the various segments of the body rotated with the same angular velocity. This is clearly not the case when the thighs and lower legs are moving toward the trunk as the diver assumes the tuck position.

Example 4.8. Gymnast on Rings. The feet of a gymnast of mass 2m and height 2L are attached to two rings as shown in Fig. 4.11. The gymnast is let go from rest in the horizontal position as indicated in the figure. The gymnast swings down keeping her body aligned in a straight line. To assess the loads carried by her abdominal and back muscles during the swing, let us model the gymnast with two rods (OA and AB) connected by a hinge at point A. The point O represents the feet, and we assume it to be stationary in the reference frame E. When these two rods let go from rest in the horizontal configuration, will they begin to rotate as one solid body? Does the gymnast contract back (or abdominal) muscles to remain straight? What are the angular accelerations of rods OA and AB?

Solution: Free-body diagrams of rods OA and AB shown in Fig. 4.11 indicate that there are six unknowns in this problem: the horizontal and vertical reaction forces exerted by the hinge (rings) on the rod OA, the horizontal and vertical forces OA exerts on AB, and the angular accelerations OA and AB. Newton's third law dictates that the resultant force AB exerts on OA must be equal in magnitude but opposite in direction of the force OA exerts on AB. We use the equations of motion for the center of mass of each rod (points D and E in Fig. 4.11c, lower diagrams) and the conservation of angular momentum to evaluate these unknowns.