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 (velocity). The other end of the thread 

 is fixed and the particle is held at a distance b (>a) below the fixed point. 

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

 n 2 (b-a)> or <g, (ii) in its subsequent motion it will oscillate about a 

 point which is at a distance a+g/n 2 below the fixed point, (iii) the distances 

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

 (iv) the interval between any two positions of rest is TT/WI, where m 2 =n 2 K 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 = f(v)cos<f>, 



where < is the angle the direction 



-j- of motion at time t makes with the 



Fig. 54. horizontal and u is the horizontal 



velocity, so that u = v cos <f>. 



Again resolving along the normal to path, since the resistance 

 is directed along the tangent, we have 



= g cos </>, 



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



v$=g cos <f>, 

 and thus, eliminating t, 



du vf(v) , 



-77 = , where v u sec 6. 



d(f> g 



