81, 82] 



17) 



CENTRIFUGAL COUPLE. 

 H x L e + H y M c + H 2 N = 0, 



257 



it is also perpendicular to the angular momentum. Consequently 

 the axis of the centrifugal couple 

 is perpendicular to the plane 

 containing the instantaneous axis 

 and the axis of angular momen- 

 tum, and is drawn in such a 

 direction that if 8 C be turned 

 through a right angle in the 

 direction of the body's motion it 

 will coincide in direction with H 2 . 

 Also since the components of S c 

 are equal in absolute value to the 

 components of H% multiplied 

 by co, we have 



Fig. 75. 



18) 



S c = HZ o = oH sin (Hci). 



Thus the centrifugal couple is equal to the vector product of 

 the angular momentum by the angular velocity. In case the instan- 

 taneous axis is a principal axis at 0, the direction of H coincides 

 with that of and the centrifugal couple vanishes. The body will 

 then remain permanently turning about the same axis. This property 

 of a heavy body turning about its center of gravity about a principal 

 axis of maintaining the direction of that axis fixed in space was 

 utilized by Foucault in his gyroscope, the axis of which points in a 

 fixed direction while the earth turns, and thus the motion of the 

 earth is made observable. The same principle is utilized practically 

 in the Obry steering gear contained in the Whitehead automobile 

 torpedo, in which a rapidly rotating gyroscope is made to give the 

 direction to the torpedo, and by acting on the steering gear to make 

 it return to its course if it accidentally leaves it. 



Suppose on the other hand that the instantaneous axis is not a 

 principal axis. The centrifugal couple then tends to generate an 

 angular momentum whose axis is in its own direction, and this new 

 momentum compounds with that which the body already possesses. 

 Let us consider two successive positions of the body. Suppose that 

 in the time dt the body turns about the instantaneous axis through 

 an angle d(p = &dt. At the end of that time the vector H 2 (Fig. 75), 

 would have turned through the angle dcp into the position H 2 , the 

 length of the infinitesimal vector H 2 H 2 being H 2 dcp = H 2 &dt. But 

 during this time the centrifugal couple S c has given rise to the 

 angular momentum 



/ 



WEBSTER, Dynamics. 



17 



