48 MOTION OF A SPINNING BODY
angular speed of each particle about the axis JD decreases when the particle is receding from this axis. The angular motion about the axis JD of each particle in the quadrant ODG is decelerating away from the reader (accelerating toward the reader), while the motion of each particle in the quadrant OJA is accelerating away from the reader. Thus, each particle in the quadrant ODG is acted upon by an external force toward the reader, while each particle in the quadrant OJA is acted upon by a force away from the reader.
The resultant of the forces acting on all the particles in the quadrant OAD is represented by the line F,, Fig. 37. The resultants of the forces acting on the particles in each of the other three quadrants are represented by the lines of equal length, F2, F3 and F4, respectively. These forces tend to turn the spin-axle about an axis AG in the direction indicated by the arrow-head marked wp.
It has now been shown that whenever an external torque is applied to a symmetrical unconstrained spinning body, the spin-axis tends to become parallel to the torque-axis, and with the direction of spin in the direction of the torque.
If the unconstrained spinning body be acted upon simultaneously by more than one torque, then the spin-axis tends to set itself parallel to the resultant torque.
In the case of a wheel mounted so that there is a fixed angle between the spin-axis and the torque-axis, the torque-axis retreats from the advancing spin-axis. The spin-axis continues to move in the direction to make it become parallel to the torque-axis, but it cannot become parallel on account of the rigidity of the connection between the wheel and its supporting frame.
31. Precession. - In the preceding Article it has been shown that when a symmetrical unconstrained spinning body is acted upon by an external torque about an axis perpendicular to the spin-axle, the spin-axle will rotate about an axis perpendicular to the axes of spin and of torque. For example, with the directions of spin and of external torque as indicated by the arrows labeled hs and L in Fig. 38, there would be a rotation of the spin-axle about the axis AG in the direction indicated by the arrow labeled wp. It will now be shown that this rotation of the spin-axle develops an internal torque that is in opposition to the applied external torque.
Because of the angular velocity wp, a particle in the semicircle ADGO is approaching the reader, and a particle in the semicircle GJAO is receding from the reader. A particle at either A or G has zero linear velocity and maximum linear acceleration perpen