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



4. A particle of unit mass is fastened to one end of an elastic thread 

 of natural length a and modulus an 2 , in a medium the resistance of which 

 to the motion of the particle is 2/c (velocity). The other end of the thread 

 is fixed and the particle is held at a distance b (&amp;gt;a) below the fixed point. 

 Prove that, when set free, (i) it will begin to rise or fall according as 

 n 2 (b-a)&amp;gt; or &amp;lt;g, (ii) in its subsequent motion it will oscillate about a 

 point which is at a distance a+g/ri* below the fixed point, (iii) the distances 

 from of successive positions of rest form a geometric series of ratio e -*/, 

 (i v) the interval between any two positions of rest is n/m, where m 2 = ri* * 2 . 



5. A particle moves on a smooth cycloid whose axis is vertical and vertex 

 downwards under gravity and a resistance varying as the velocity. Prove that 

 the time of falling from any point to the vertex is independent of the starting 

 point. 



6. A particle moves under a central force $ (r) in a medium of which the 

 resistance varies as the velocity. Investigate the equations 



where h and p, are constants. 



*212. Motion in a vertical plane under gravity. For 



any law of resistance we can make some progress with the equa 

 tions of motion of a particle moving in a vertical plane under 

 gravity. 



Let mf(v) be the magnitude of the resistance when the 



velocity is v, m being the mass of 

 the particle, then resolving hori 

 zontally we have 



u = /(v)cos&amp;lt;/&amp;gt;, 



where &amp;lt;/&amp;gt; is the angle .the direction 



of motion at time & makes with the 

 Fig. 54. horizontal and u is the horizontal 



velocity, so that u = v cos &amp;lt;f&amp;gt;. 



Again resolving along the normal to path, since the resistance 

 is directed along the tangent, we have 



v 2 



= g COS 9, 



where p is the radius of curvature. This equation may be written 



vfy = g cos &amp;lt;f&amp;gt;, 

 and thus, eliminating t, 



du vf(v) , , 



= J ^ / . where v = u sec 6. 

 d$ g 



