34 KINETICS OF A PARTICLE. [62. 



(a) Determine the velocity v of P at any distance BP = s, and its 

 velocity v 1 at B. (b} How is the solution to be modified if the linear 

 mass BA extends from B to infinity ? 



(8) A circular wire of radius a and constant density p attracts, 

 according to Newton's law, a particle P of mass i, situated on the axis 

 of the circle ; i.e. on the perpendicular to its plane passing through the 

 centre O. If the velocity is zero when the particle is at the distance 

 OP^ = s , determine the velocity of the particle at any distance s, and 

 show that the motion is oscillatory. 



(9) Determine the motion of two free particles of masses m lf m 2f . 

 attracting each other according to Newton's law, and starting at the 

 distance j with zero velocity. 



(10) Show that the motion of the particle in Art. 61 is oscillatory, 

 and that the period, i.e. the time of one complete oscillation, is 



(n) A particle of mass m is suspended from a fixed point by means 

 of an elastic string whose weight is neglected. The natural length of 

 the string is /. Its length, when the mass m is suspended at its end, is 

 /! If the particle be pulled down so as to make the length of the 

 string = s , and then released, the particle will perform oscillations. 

 Determine their period : (a) if s / x < / x /; () if s ^ > ^ I. 



(12) The particle in Ex. (u) is raised through a height h, so as to 

 loosen the string, and then dropped. Determine the greatest exten- 

 sion of the string. 



(13) An elastic string, whose natural length is = /, is suspended 

 from a fixed point. A mass m attached to its lower end stretches it to 

 a length / x ; another mass m 2 stretches it to a length / 2 . If both these 

 masses be attached and then the mass m 2 be cut off, what will be the 

 motion of m l ? 



(14) A particle performs rectilinear oscillations owing to a centre of 

 force in the line of motion attracting the particle with a force directly 

 proportional to the distance. The motion of the particle is impeded 

 by a resistance directly proportional to the velocity. Investigate the 

 motion. 



