Velocity and Acceleration in Polar Coordinates

During an arm wrestle, the forearm of the man who is at the brink of defeat will begin to draw a circle whose center is his elbow pushed against a table. In this case, in the computation of velocity and acceleration of the forearm, it may be easier to use polar coordinates rather than Cartesian coordinates.

In polar coordinates, we define er to be the unit vector in the direction of the position vector connecting origin O of the coordinate system to a moving point P. Consider, for example, the case of abduction of the arm as shown in Fig. 2.7a. The unit vector along the line of the arm er is given by the equation:

Then the position vector connecting the shoulder to the elbow can be written as r = L er

Figure 2.7a,b. Polar and path coordinates. The unit vectors associated with polar coordinates are er and et. The vector er is in the radial direction pointing outward whereas et is tangent to the circle and points in the direction of increasing 6 (a). In the case of path coordinates, the unit vector n is normal and t is tangent to the trajectory (b). The symbol p denotes the radius of curvature; it is the radius of the largest circle that is tangent to the particle path at the location shown.

Figure 2.7a,b. Polar and path coordinates. The unit vectors associated with polar coordinates are er and et. The vector er is in the radial direction pointing outward whereas et is tangent to the circle and points in the direction of increasing 6 (a). In the case of path coordinates, the unit vector n is normal and t is tangent to the trajectory (b). The symbol p denotes the radius of curvature; it is the radius of the largest circle that is tangent to the particle path at the location shown.

in which L denotes the length of the upper arm. Taking the time derivative of the position vector, we determine the velocity of the elbow:

= L (d6/dt) et in which et is perpendicular to er as shown in Fig. 2.7.

Next, let us determine acceleration by taking the time derivative of velocity v:

= L (d26/dt2) (-sin 6 e1 + cos 6 e2) - L (d6/dt)2 (cos 6 e1 + sin 6 e2)

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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