ROTATION AND HORIZONTAL MOTIONS BRASCHMANN 



29 



velocity of this point, then 



dx 

 dt 



dy 

 dt 



= v sin a 



v cos a 



Let v x v v z be the components of the momentary velocity along 

 the axes x y z and hence their differential quotients with respect 

 to t will be the momentary accelerations in these directions and 

 identical with 



d 2 x d 2 y d 2 z 



dt 2 dt 2 



Equation (2 X ) now becomes 

 d 2 x x _ dv 3 

 ~d~f It 



d? 



2 co sin X.v . cos a 



d 2 y, dv„ , _ . , 

 ■ /1 = y + 2 co sin / . v . sin a > 



d* 2 



d 2 ^ 



d< 





+ 2 to cos ^ . v . sin a 



(4) 



Now gravity and friction are the accelerative forces acting on a 

 point in contact with the sides of the track or path of constraint. 

 The projections of gravity on the horizontal plane are equal to 

 zero and equally so the projections of the lateral friction on the 

 direction of the lateral pressure disappear. Hence if we designate 

 the acceleration of gravity by g we have the forces 



X = 



Y = 



Z=g 



and equation (4) gives the pressure 



dv 



P-- 



dt 



+ 2 to sin X.v . cos a 



P = — — 2 — 2 co sin yLv.sin a > (5) 



y dt 



P, = e — — 2 co cos A.v. sin a 

 2 * dt 



