## B

Figure 4.8a,b. Examples of rigid objects that have planar symmetry (a) and those that do not have any plane of symmetry (b).

Figure 4.8a,b. Examples of rigid objects that have planar symmetry (a) and those that do not have any plane of symmetry (b).

Mass moment of inertia with respect to any other point, say point O, can be found by using the geometric relation:

where m is the mass of the object and r is the distance between O and C. This relation is called the parallel axis theorem. Earlier in the chapter, we found that the mass moment of inertia of a slender rod with respect to one of its ends was given by the following relation:

Using Eqn. 4.27, we see that the mass moment of inertia with respect to the mass center Ic = mL2/12 for the slender rod considered here.

Mass moment of inertia is an entity that readily applies to three-dimensional geometries, as we shall see later in the book. In three dimensions, mass moment of inertia with respect to center of mass has nine e e

e components that can be organized into a 3 X 3 symmetrical matrix. That means that of the nine components only six are independent; I23 = —132, and so on.

Using Eqns. 4.22, 4.24, and 4.27a, the moment of momentum of a rigid body B about a material point of the rigid body can be written as

HO = rC/O X m vO + a (I13 e1 + I23 e2 + I33 e3) (4.27b)

where O is a point in the rigid body and m is its mass. If O is a point that is fixed in a reference frame which is stationary with respect to earth, then vO = 0. Furthermore, if the plane of motion coincides with a plane of symmetry of the rigid body, then I13 = I23 = 0.

One final comment about the moment of momentum of a rigid body: because the rate of rotation of all points in a rigid body is the same, moment of momentum is called angular momentum.

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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.

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