Three-dimensional human movement plays a fundamental role in performing arts and athletics. Some of us may have observed the three-dimensional nature of baseball pitching or noticed how a diver can induce twisting rotations in air by raising an arm to the side. Appropriately chosen shape changes are crucial in gymnastics, in the graceful turns of a ballerina, and the triple and quadruple jumps in ice skating. In this chapter, we present the principles of three-dimensional mechanics of bodies and their application to human motion. The mechanics of rigid bodies was laid out in 1760 by the Swiss scientist Leonhard Euler in his book Thoria Motus Corporum Solidurum seu Rigidorum. Euler is considered to be a founder of pure mathematics. He was a profilic scholar. Reading and writing late into the night under a gas lamp, night after night, he lost the sight of one eye at the age of 28. He became totally blind at 60. Blindness did not diminish his productivity, however. He had a unique ability in carrying out mental computations and continued to write papers and books for another 15 years, until he died in 1783.

Euler treated a rigid body as a system of particles in which the distance between any two particles remained constant with time. Using this property, Euler derived equations that govern the rate of rotation of a rigid body in three dimensions. These equations are much more complex than those that govern the planar motion of rigid bodies. The simple definition of angular velocity in planar motion no longer applies to three-dimensional motion. The derivation of equations of motion in three dimensions involves seemingly abstract concepts concerning vector differentiation. In the first three subsections of this chapter, we outline the application of Newton's laws of motion to the three-dimensional motion of rigid objects and multibody systems. Our presentation in this chapter does not closely follow that of Euler. Especially in the description of three-

dimensional motion, we follow the presentation in Thomas R. Kane's Dynamics.

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Getting Started With Dumbbells

Getting Started With Dumbbells

The use of dumbbells gives you a much more comprehensive strengthening effect because the workout engages your stabilizer muscles, in addition to the muscle you may be pin-pointing. Without all of the belts and artificial stabilizers of a machine, you also engage your core muscles, which are your body's natural stabilizers.

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