Laws of motion for a body in three-dimensional motion are as follows:
2F = m ac 2Mc = EdHc/ dt where 2F is the resultant external force acting on the body, 2Mc is the resultant external moment with respect to the center of mass, and ac is the acceleration of the center of mass measured in the inertial reference frame E. The term Hc represents the moment of momentum about the center of mass.
If a point within the body is fixed in the inertial reference frame E, then the following equation also holds
2Mo = EdHo/dt where 2Mo and Ho represent the resultant moment and the moment of momentum about O.
The moment of momentum of a rigid body is called angular momentum. For three-dimensional motion, one obtains the following expression for angular momentum:
in which Hci and «i are the components of angular momentum and angular velocity of the rigid body in reference frame E written in some ref erence frame B. Terms Icj are elements of mass moment of inertia. These elements depend only on the geometry and mass density distribution of the rigid body.
The angular momentum about the fixed point O has a similar expression:
Ho1 = (Io11 «1 + Io12 «2 + Io13 «3) Ho2 = (Io21 «1 + Io22 «2 + Io23 «3)
As shown in the chapter, once the inertia matrix Icj is derived, the matrix Ioj can be obtained from the matrix Icj by using a transformation equation (Eqn. 9.17).
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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.