12 



I. KINEMATICS OF A POINT. LAWS OF MOTION. 



that is, 



22) 



dt } 



which might have been obtained from the expression for the lengths 

 of the arc in polar coordinates. 



8. Sector Velocity. Let the polar coordinates of a point 

 at -the time t be r, qp, and let the area of the sector enclosed between 



the path, the fixed line of reference, 

 and the radius vector be denoted by S. 

 If denote the angle made by the 

 tangent to the path, in the direction 

 of motion with the direction of the 

 radius vector from the origin, we 

 have (Fig. 5) 



dr = ds cos e, 

 rd<p = ds sin , 



and if in the time dt the area of the 

 sector increases by dS, we have 



Fig. 5. 



dS = -^ rds sin s = r 2 d(p. 



The rate of increase of the area of the sector 



23) 



~dt 



may be called the sector velocity, and making use of the value 



we see that it is equal to one-half the product of the 

 ds 



2 dt 



magnitude of the velocity -^- and the perpendicular distance, d = r sin s 



from the origin to the line of direction of the velocity, that is, to 

 one -half the moment of the velocity. Therefore the sector velocity 

 may be represented by a vector perpendicular to the plane OAB, 

 the components of which will be 



24) 



dSy 



dt 



and we also have 

 25) 



dt 



- (*!\i (*8gY, (*8,\ 

 ~ \dt ) " h \dt)~ r \dt ) 



