## Rolling of an Abdominal Wheel on a Horizontal Plane

In this section, we consider the three-dimensional motion of an abdominal wheel on a flat surface. The solution is obtained by using the definition of angular momentum directly. Later we explain how the problem could be solved by expressing angular momentum in terms of the mass moment of inertia and angular velocity.

Example 9.7. Abdominal Wheel. A simple mechanical device called an abdominal wheel was promoted in the 1980s as the miracle tool for strengthening abdominal muscles. It is composed of a circular disk with two short cylindrical bars on both sides (holders) (Fig. 9.10). The radius of the disk is R and the length of each of the holders is L. When placed on a flat floor, the disk of the abdominal wheel rolls without slip in such a way that the center C of the disk travels on a horizontal circle at constant velocity vo. The center of this circle is the distal end of one of the holders. The disk has a mass of m and the weight of the holders is negligible. Determine the angular velocity of the rotating disk in the inertial reference frame E. Determine also the ground forces acting on the abdominal wheel.  V Oe

V Oe A

Figure 9.10a-d. The motion of an abdominal wheel on a horizontal plane (a). The abdominal wheel is composed of a disk of mass m and two rods attached to it. These rods are used to hold the wheel and push it out on the floor and away from the body. The weights of these rods are small compared to the weight of the disk. When set on the floor with a velocity, the abdominal wheel tends to rotate around one end of one of the holders. The reference frames used in the analysis are identified in (b) and (c). The free-body diagram of the abdominal wheel is given in (d). 