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



62. A cycloidal tube, of which the radius of the generating circle is a, is 

 placed with axis vertical and vertex downwards, and contains two elastic 

 threads of natural length I fastened at one end to the extremities of the base 

 and at their other ends to a particle. If the particle is moved a distance 

 x from the vertex, where x&amp;lt;4a-l, it will reach the vertex in time 



8na+l; 



where n is the ratio of the modulus of the string to the weight of the particle. 

 Find the time taken if x&amp;gt;a l. 



63. Two cycloids are placed in the same vertical plane, with their axes 

 vertical, and their vertices downwards and at the same level. Two particles 

 start to describe the cycloids from points at the same level. Show that they 

 will next be at the same level after a time 2?r *J(aa )l{(*Ja + *Jaf) v /^}, and next 

 after that at time 4?r J(aa )l{(Ja + Ja ) Jg] or 2?r v/(aa )/{(\/~V a ) *fc} which 

 ever is less, a and a being the radii of the generating circles. 



64. An endless thread of length 2Z, on which are threaded beads of masses 

 M and m, passes over two small smooth pegs A and B, which are at a distance 

 a apart and in a horizontal line. The lighter bead m is raised to the middle 

 point of AB and is then let go. Show that the beads will just meet if 



65. Two particles A, B are connected by a thread of length I which 

 passes through a small hole C in a smooth horizontal table on which A moves 

 and supports B. A is projected along the table at right angles to AC. Show 

 that, if AC=&amp;lt;lj and if n is the ratio of the masses of B and A, B cannot 

 reach the table if the velocity of projection is less than that due to falling 

 through a height W/(I + K). 



66. Two particles of masses m and KM are connected by a thread which 

 passes over the top of a smooth circle, the particles lying on the circle. Show 

 that the motion of m from its position of equilibrium will b,e the same as that 

 of a free particle starting from the top of the circle, under gravity diminished 

 in the ratio x /(l + K 2 + 2*cosa) : I + K, a being the angle the connecting thread 

 subtends at the centre. 



67. A straight smooth groove is cut in a horizontal table, and a straight 

 slit is cut in the bottom of the groove. A thread of length I attached at one 

 end to a particle of mass m resting in the groove passes through the slit and 

 supports a particle of mass Km. The second particle is held displaced in the 

 vertical plane containing the slit with the string straight, and is let go. 

 Prove that its path is part of an ellipse of semi-axes I, and 11(1 + *c), the major 

 axis being vertical. 



68. Two particles A, B each of mass m slide on a circular wire of radius a 

 fixed in a vertical plane, and are connected with a third particle C of mass m! 

 by two threads each equal to the radius. The system starts from rest in a 



