CHAPTER X. 



MOTION UNDER CONSTRAINTS AND RESISTANCES. 



182. THE second main subdivision of &quot;Dynamics of a Particle&quot; 

 relates to motion of a particle in a given field of force when the 

 force of the field is not the only force acting on the particle, but 

 there are unknown forces also acting upon it and enforcing some 

 conditions. This subdivision includes all cases where the particle is 

 constrained to move on a given curve or surface smooth or rough, 

 all cases where the motion of one particle is partly determined by 

 that of another with which it is connected in an invariable manner, 

 and we shall extend it to include all cases of particles subject to 

 motional forces (Article 152). 



183. Motion on a smooth curve in a vertical plane 

 under gravity. The motion is entirely determined by the equa 

 tion of energy 



i^ 2 + gy const., 



where v is the velocity of the particle when it is at a height y 

 above a fixed horizontal plane. The velocity is therefore always 

 that due to falling from a definite level which is the same through 

 out the motion. 



Let p be the radius of curvature at the point where the velocity 

 is v, and &amp;lt;f&amp;gt; the angle which the normal drawn inwards (towards 

 the centre of curvature) makes with the vertical drawn downwards, 

 then, taking m for the mass of the particle and R for the pressure 

 of the curve on the particle, we have by resolving along the normal 



v 2 



m = mg cos &amp;lt;/&amp;gt; + R, 

 P 



where R is supposed to act inwards. 



