## Velocity and Acceleration in Path Coordinates

Even when a particle draws a planar curved path that is not a circle, it is possible to describe its velocity and acceleration in terms of unit vectors that are tangential and normal to the path. Consider a particle traversing a complex planar curve. Let t be the unit vector tangent to the particle path and let n be the unit normal vector drawn outward as shown in Fig. 2.7b. The velocity and acceleration of particle P can then be written as v = v t = (ds/dt) t (2.10a)

in which s is the arc length along the particle path, v = ds/dt is the speed of the particle, and p is the radius of curvature. it is defined as the radius of the largest circle that has the same tangent with the particle path at point P. In a sense, Eqn. 2.10 can be considered as a generalization of Eqn. 2.9. It is used quite often in the analysis of motion of satellites and stars because of the elliptical nature of their particle path.

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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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