Angular Velocity

Let b1, b2, b3 be a right-handed set of mutually perpendicular unit vectors fixed in a rigid body B. The angular velocity R«B in a reference frame R is defined as

RWB = [(Rdb1/di) • b2]b3 + [(Rdb2/di) • b3]b1 + [(Rdb3/di) • b1]b2

where (Rdbi/di) denotes the ordinary derivative of bi with respect to time in reference frame R. This definition of angular velocity leads to the following results:

Rdb1/di = R«B x b1 Rdb2/di = R«B x b2 Rdb3/di = R«B x b3

Thus, for any vector P defined in the reference frames R and B:

If a series of succesive reference frames is used in the study of motion, the following equation holds:

Velocity and acceleration of two points in a rigid body are related by the following set of equations:

where EaB is the angular acceleration of the rigid body B. The angular acceleration is defined by the relation

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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