112 KINETICS OF A PARTICLE. [208. 



Their integration can best be performed by introducing the radius 

 vector r by means of the relations (22). Multiplying the equations 

 respectively by cosw/ and sinco/, and adding, we find that the right- 

 hand member vanishes, and we have 



x cos w/ + y sin w/ == o, 

 or, substituting for x and y their values from (22), 



r w?r = o, 

 which agrees with (23), Art. 205. 



208. For the pressure on the curve, we have, by ( 18), since <f> s ?-t-<j>f= i, 

 N= \ = _^!L (x s in <o/ + j> cos o>/). 



COS 2 CO/ 



Substituting from (22) and (24), and reducing, we find 



N= ///a>[ (oo + z>o)^(i + tan 2 o>/) + (o>r V Q ) <?-"*( i tan 2 co/)]. 



209. Exercises.* 



(1) A particle subject to gravity moves without friction in a straight 

 tube which revolves uniformly in a vertical circle. Find the distance r 

 of the particle from the centre of rotation at any time /. 



(2) A particle moves without friction in a circular tube which rotates 

 uniformly in a horizontal plane about a point O in its circumference. 

 If the particle is at the time /=o at rest at the end of the diameter 

 passing through O, what is its position at any time / ? 



(3) A particle moves in a horizontal circular tube whose radius 

 increases proportionally to the time. At the time / = o the radius is a, 

 and the particle has a velocity V Q perpendicular to the radius. Find the 

 position and velocity at any time /. 



* These examples are taken from Walton's Collection (referred to in Art. 159), 

 pp. 401-406. 



