MOTION OF A SPINNING BODY 61
the axis of rotation of w are unaffected by the rotation about that axis. All particles of the wheel are moving about the spin-axis h,. While a particle of the wheel moves from the point in space A to the point B, the component linear velocity of the particle changes in the direction of the spin-axis from a finite value to zero. Suppose that the points A, b, c, d, B, Fig. 47, are the positions of a particle after moving for equal intervals of time with constant speed about the spin-axle perpendicular to the plane of the diagram. If the plane of the wheel is at the same time rotating about an axis DB, the change in the magnitude of the linear velocity in the direction perpendicular to the plane of the wheel of a particle while moving from A to b is represented by Aa', and the changes while moving during succeeding equal intervals of time are represented by bb', cc', dd'. The rate of change of linear velocity increases as the particle moves from A to B. The inertia of the particle causes it to oppose this change of velocity by a force proportional to the rate of change of linear velocity of the particle. The forces exerted by the particle when at various points between A and B in opposition to the change of linear velocity normal to the plane of the wheel, are represented by arrows in Fig. 46.
While moving from B to C, the component of the velocity of the particle normal to the plane of the wheel increases from zero at B to a maximum at C. During the displacement from B to C, the inertia of the particle causes it to oppose this change of velocity at each point of its path by a force which is in the direction opposite to the velocity normal to the plane of the wheel. In Fig. 46, the forces at different points from B to C are represented by arrows. It is thus seen that each particle in the half of the wheel ABC tends to rise, thereby producing a rotation of the wheel about the torque-axis L. In a similar manner it can be shown that the half of the wheel CDA tends to rotate downward about the same axis.
Therefore, when the axle of a' spinning body is rotating about an axis perpendicular to the spin-axis, a torque is developed on the body about an axis perpendicular to the plane of the spin-axis and the axis of rotation. The direction of this internal torque is opposite that of the external torque which would need to be applied about the same axis in order to produce a precession of the gyro-axle in the direction in which the gyro-axis is rotating.
The reaction of this torque acts in the opposite direction about the same axis on the agent that causes the rotation of the spin-axle.
When a torque is applied to an unspinning gyro about an axis