To relate the velocity of two points in a rigid body, one takes the time derivative of the position vector connecting the two points:
Subsequent differentiation of this relation with respect to time yields the following equation relating accelerations of points P and Q:
EaP = EaQ + EaB X rP/Q + E«B X (E«B X rP/Q) (9.10b)
where EaB is the angular acceleration of the rigid body B. The angular acceleration is defined as the time derivative of angular velocity:
Equations 9.10 and 9.11 are identical to those presented earlier for planar motion. It is just that the definition of angular velocity is more complex in three-dimensional motion than in planar motion.
Example 9.2. Leg-Lifting Turns of a Dancer. A dancer rotates his body around an axis normal to the stage at a constant rate of 2 rad/s, counterclockwise (Fig. 9.5). At the same time, his one leg moves away from his body axis (extends) at a uniform rate of 10 rad/s. Determine the an-
gular velocity of the dancer's trunk as well as the angular velocity of his leg with respect to his trunk.
Solution: Let bi and ei represent the unit vectors attached to the trunk of the dancer and to the floor (Fig. 9.5). The angular velocity of the trunk with respect to the inertial reference frame E is given by the following equation:
Similarly, we can show that
where D denotes the reference frame attached to the extended leg of the dancer. The angular velocity of the leg of the dancer with respect to earth is given by the equation:
E«D = BwD + EwB = -10 (-sin 0 e1 + cos 0 e2) + 2e3 = -10b2 + 2e3
Example 9.3. Velocity and Acceleration During a Leg-Lifting Turn. Compute the velocity and acceleration of the hip and the ankle of the dancer performing a leg-lifting turn. The velocity of the center of mass C at the moment considered was equal to zero. The distance between the two hip joints of the dancer was measured to be 22 cm. The total length of lower limb was 88 cm. At the instant considered the leg was aligned with the negative e3 axis. All other parameters were the same as in Example 9.2.
Solution: Let E, B, and D represent the reference frames fixed on the stage, on his trunk, and on his leg, respectively. Let Q denote the rotation center of the hip joint. Using the angular velocity that was determined in the previous example, we find that:
EvQ = EvC + EwB X rQ/C = 0 + 2e3 X (0.11b2) = -0.22bi
Let us now use Eqn. 9.10 between the hip joint (Q) and the center of rotation of the ankle (P) to determine the velocity and acceleration of his ankle.
To compute the acceleration of P, we need to determine the angular acceleration EaD of the leg with respect to E.
Now we can use Eqn. 9.10b to compute EaP:
Because the leg is oriented in the direction of e3, we find:
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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.