An example of planar motion of a body that has no symmetry with respect to a plane parallel to the plane of motion is considered. We find that angular momentum is not perpendicular to the plane of motion. Furthermore, contact forces that support the planar motion are found to be three-dimensional.
Example 9.6. Dancing Around a Pole. Consider a dancer rotating around a pole while holding onto it (Fig. 9.9). The dancer's feet are placed next to the pole and she skips in place as she rotates around the pole at a constant rate «o. The angle of inclination 0 of the dancer with the vertical axis remains constant at all times. Our aim is to determine the reaction forces exerted on the dancer as a function of angular velocity, height, and weight. These forces consist of the ground forces exerted on the feet and the force exerted by the pole on the hands of the dancer.
Solution: We represent the dancer as a slender rod with length L and mass m (Fig. 9.9b). We seek to determine the angular velocity of the slender rod. We denote the unit vector along the line of the rod b3. Let n and t be auxiliary unit vectors in the horizontal plane (e1, e2) defined in Fig.
9.9b. Note that t is in the direction of the velocity of the center of mass. Let bi be the unit vector normal to b3 in the plane of b3 and n. The unit vector b2 is then perpendicular to both b1 and b3:
n = cos $ e1 + sin $ e2 t = —sin $ e1 + cos $ e2 b1 = cos 0 (—cos $ e1 — sin $ e2) + sin 0 e3 (9.25a)
b2 = —t = sin $ e1 — cos $ e2 b3 = sin 0 (cos $ e1 + sin $ e2) + cos 0 e3
The unit vectors b1x b2, and b3 comprise a set of orthogonal unit vectors embedded in the slender bar B. We call this the reference system B. The angular velocity of the rod in the inertial frame E can then be obtained by using the three-dimensional definition of angular velocity given in Eqn. 9.5. Note that the angle 6 remains constant during motion and therefore its time derivative is zero. Note also that the unit vectors e1, e2, and e3 are also constant in E. Thus we obtain:
Substituting these expressions into Eqn. 9.5 yields the result:
We could have arrived at this expression without the use of Eqn. 9.5 by recognizing that a line drawn in the vertical direction onto a dancer never changes its orientation during the dancer's motion. Thus the planar definition of angular velocity, Eqn. 9.3, holds and once again m = wo e3.
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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.