184 MOTION UNDER CONSTRAINTS AND RESISTANCES. [CHAP. X. 



of the value of the work function in the position at time t above 

 its value in the standard position. 



The integral equation gives the velocity v in each position, 

 and the second of the equations of motion gives the pressure R. 

 In the case of one-sided constraint the particle leaves the curve 

 when R changes sign. 



196. Examples. 



1. Prove that, when the particle leaves the curve, the velocity is that 

 due to falling under the force kept constant through one quarter of the chord 

 of curvature in the direction of the force. 



2. Prove that, when the curve is a free path under the given forces for 

 proper velocity of projection, then for any other velocity of projection, the 

 pressure varies as the curvature. 



*197. Smooth plane tube rotating in its plane. Suppose 



that a particle of mass ra moves 

 in a smooth plane tube, and that 

 the tube rotates in its plane about 

 a point rigidly connected with 

 it. Let OA be any particular 

 radius vector of the tube, and 

 <f) the angle OA makes with a 

 fixed line in the plane of the 

 tube. Then < is the angular 

 velocity of the tube. We shall 

 Fig. 50. write co for cj>. 



Let P be the position of the particle in the tube at time t. 

 Let OP = r, and Z A OP = 0. Then r and 6 are polar coordinates 

 of P referred to OA as initial line, and r and + <f> are polar 

 coordinates of P referred to a fixed initial line. Let p be the 

 radius of curvature of the tube at P. 



Let v be the velocity of the particle relative to the tube. 

 Then, if arc AP = s, v is s, the direction of v is that of the tangent 

 to the tube, and the resolved parts of v along OP and at right 

 angles to OP are f and r6. 



Now the resolved accelerations of the particle along OP and at 

 right angles to OP are 



