14 



I. KINEMATICS OF A POINT. LAWS OF MOTION. 



This is not in general equal to -^y which is the acceleration of 



the scalar velocity. The direction of a is given by its direction 

 cosines , 



27) cos (ax) = TTJ-/ a, cos (ay) = jiir a> 9 cos (as) = 



1O. Acceleration Components. We may now find the com- 

 ponent of the acceleration in the direction of the tangent to the path. 

 The direction cosines of the tangent being, by 6, 



* v y v * 



; J } 



V V V 



we have for the tangential component by 9) 



rtl rtl A* 



28) 



a* = a x ^ 4. a JL 4. a _^ 

 v y v v 



_ I \dx d*x dy d*y 

 ~ ~v(dt ~dW ' ~dt ~di r 

 But differentiating equation 17) 



dz 

 dt 



dv 



ds d z s dx d*x dy 



^7^ ^7* ^7*2 Jj. J*2 I Jj. Jj.2 i J 



dz d 2 



4t 



* 



<dt* 



and dividing by v and comparing with 28) we find 



that is the acceleration of the scalar velocity is the projection of the 



vector acceleration on the tangent. This is called the tangential 



acceleration, or acceleration 

 in the path. 



We may obtain a con- 

 venient expression for the 

 remaining component of the 

 acceleration. If P and Q (Fig. 7) 

 be two "consecutive" points 

 of the path, the plane con- 

 taining the tangents at P 

 and Q is called the osculating 

 plane, or plane of principal 

 curvature. Normals drawn 

 in this plane are called 

 principal normals, and the 

 point where they intersect, 

 the center of curvature. The 

 radius OP=$ is called the 

 radius of curvature. If the 



angle between the consecutive tangents is AT and the distance between 



the points P, Q is As, the curvature is defined as 



