DYNAMICS OF A POINT, 

 which intersects it, and a particle is projected from the 



stationary point of the tube with a velocity ^ ; find its 



w^sma) 



position at any time before it attains relative equilibrium; and 

 pn>ve that the equilibrium is unstable. 



1431. A smooth parabolic tube of latus rectum I is made to 

 v-e about its axis, which is vertical, with angular velocity 



/j, and a particle is projected from the vertex up the tube; 



prove that the velocity of the particle is constant, and that the 

 greatest height to which the particle rises in the tube is twice 

 that due to the velocity of projection. 



1 I'M. A. smooth parabolic tube revolves with uniform angular 

 velocity about its axis, which is vertical, and a particle is placed 

 within the tube very near to the lowest point : find the least 

 angular velocity which the tube can have in order that the 

 particle may ascend; and, if it ascend, prove that its velocity will 

 be proportional to its distance from the axis. Prove also, that if 

 any position of tin- partido in the tube be one of relative equili- 

 brium, every position will be such. 



1 l.>3. A curved tube is revolving uniformly about a vertical 

 axis in its piano, and is symmetrical with retpeot t> that axis; 



the angular velocity is f' 1 , a being the radius of curvature at 

 \/ a 



tli- Vertex; pn.v- that the equilibrium ofl particle placed at the. 

 vertex will be stable or unstable according as the conic of closest 

 contact at the vertex is an -llip>- ..r hyperbola. 



1 |." I. A circular tube of radius a revolves uniformly about a 

 tor with angular velocity n / - , and a ] 



horn its lowest ith su -h velocity as just to reach 



tin; highest point; prove that the time of describing the first 



quadrant is 



