(f) As before, the direction of this movement is reversed by reversing either the direction of spin of the wheel or action of the force V. If both are reversed simultaneously the direction of the movement produced by the applied force is not altered.
The behaviour set forth above can be summarised in a general rule as follows:—If to a spinning wheel possessing three degrees of freedom a force be applied tending to turn the wheel about some axis X X, the actual motion produced will not be about X X but about some other axis Y Y; this axis Y Y will be such that rotation about it will tend to bring the axle of the spinning wheel into coincidence with or parallel with the axis X X; the direction of the rotation produced about Y Y will be such that when the condition of coincidence or parallelism is reached the spin of the wheel will coincide in direction with the rotation we are attempting to produce about the axis X X.
Taking case (a) ([Fig. 1]), it will be seen that the axis E F about which we are attempting to produce rotation by means of the weight W, together with the weight W itself, is of necessity carried round by the precession in the direction M at the same rate as the axle of the spinning wheel. The axle in this case cannot therefore place itself in coincidence with the axis of the applied force. But it does its best to do so. The precession persists and is an expression of the fruitless chase of the axis E F by the axle B C.
Fig. 2. Model Gyroscope, with Three Degrees of Freedom.
If, however, the weight is attached by some kind of sliding connection on the horizontal ring in such a way that its line of action remains stationary in space, then the axis about which we are attempting to produce rotation will also remain stationary in the position occupied by the axis E F before precession commences. In this case it is quite possible for the axle of the wheel to place itself in coincidence with the axis of the applied force. Precession about H J through 90 deg. will accomplish this result, as indicated in [Fig. 2]. The weight W is now acting at a point on the horizontal ring where it ceases to have any tendency to turn the wheel about the axis E F. When, therefore, the position of coincidence is reached precession ceases and the system comes to rest in this position.
If the experiment were actually made it would be found that the momentum acquired by the system during the 90 deg. turn would carry the axle through the position of coincidence with the axis of the applied force. But immediately the axle passes to the opposite side the force W is exerted on a point of the horizontal ring between F and C. The action of the force passing on to this, the opposite, segment of the ring reverses the conditions under which the system started its movement and as a result precession in the direction opposed to the arrow M is set up. The axle thus tends to recover its position of coincidence and in the end settles down to a vibratory motion from side to side of the axis of the applied weight. Friction at the vertical journals will “damp” this vibratory motion, the amplitudes of the swings will decrease, and the axle will ultimately settle in steady coincidence with the axis of the applied force. In this condition the force will have no further effect on the system beyond throwing a bending moment on to the vertical axis.
Fig. 3. Frictional Transmission of Turning Moment.
Instead of trying to make the wheel rotate about the axis E F by applying a weight to the inner ring as in [Fig. 1], let us, as shown in [Fig. 3], mount the square frame K on a horizontal axis N P and attach the weight W to an arm fixed on the frame. The axes N P and E F being—at least initially—collinear, the effect of this arrangement is to throw a turning moment on to the wheel about the axis E F just as does the weight W in [Fig. 1]. It is to be noticed, however, that the moment of the weight W in [Fig. 3] about the axis N P is transmitted to the inner ring as a moment about the axis E F solely because of the friction existing at the journals of the axis E F. This friction may be very small, so that the turning moment received by the wheel is only a very small fraction of the turning moment exerted by the weight W about N P. The effect of the arrangement is thus exactly the same as would be produced in the arrangement [Fig. 1] if we reduced the weight W to a hundredth or a thousandth of its value. In other words, precession about the vertical axis H J will set in in the direction of the arrow M just as before, but the speed of this precession will be only a hundredth or a thousandth of the previous value.