The mass elements of an arbitrarily shaped object are not aligned along a straight line as was the case for a rod, and therefore the estimation of moment of momentum requires the evaluation of a difficult integral of the vector product of velocity of a small mass element and its position vector over the volume of the object. However, because the distance between any two points in a rigid body remains constant, this integral can be reduced to a simple algebraic form. Let O denote the origin of the coordinate system E that is fixed on earth. The point C is the center of mass, and P and Q are two arbitrary points of the rigid body B undergoing planar motion parallel to the (e1, e2) plane (Fig. 4.2a). Using the vector addition property, it is clear that rQ = rP + rQ/P (4.11)
where rP and rQ are the position vectors of points P and Q and rQ/P is the vector connecting point P to point Q. As can be seen in Fig. 4.2b, this latter vector can be written as rQ/P = rQ/P [sin 6 (cos $ e1 + sin $ e2) + cos 6 e3] (4.12)
where rQ/P denotes the length of rQ/P, 6 is the angle between e3 and rQ/P, and $ is defined as the angle between e1 and the projection of rQ/P into the (e1, e2) plane.
Figure 4.2a,b. General plane motion of a rigid object B parallel to the (e1, e2) plane (a). The reference frame E is fixed on earth. The symbol C denotes the center of mass. The angles 6 and fi shown in (b) are defined in the text.
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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.