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



74. Two equal particles connected by a massless rigid rod are placed 

 in a vertical circular tube, one being at the highest point. Show that, when 

 the other reaches the lowest point, the velocity of each is the same as if they 

 had been unconnected throughout the motion. 



75. A thread of length aa has a particle attached to one extremity, while 

 the other is fastened to the highest point of a horizontal cylinder of radius a. 

 The particle is initially supported with the thread in a horizontal line at right 

 angles to the axis of the cylinder and is allowed to fall under gravity. Find 

 the tension when the thread has described a given angle, and show that, if the 

 maximum tension occurs when the thread is vertical, then a^rr. 



76. A particle is suspended by an inextensible thread of length I from a 

 point on the circumference of a cylinder of radius a whose axis is horizontal, 

 the thread being tangential to the cylinder. Prove that, if the particle is 

 projected horizontally in a plane perpendicular to the axis of the cylinder so 

 as to pass under the cylinder, the least velocity it can have in order that the 

 thread may wind itself wholly up is ^/[ {g {a/3 (1 - sin /3)}], where a/3 is the 

 length of the part to be wound up. 



77. One end of a thread of length I is attached to the highest point of a 

 fixed horizontal circular cylinder of radius a. A particle attached to the other 

 end is dropped from a position in which the thread is straight and horizontal 

 and at right angles to the axis of the cylinder. Prove that, if <27ra, the 

 thread will become slack before the particle comes to rest, and that it will 

 then have turned through an angle whose circular measure is 



78. Two particles P, Q, of equal mass, slide on a smooth endless cord 

 OPQ, which passes through a small smooth ring at 0, and lies on a smooth 

 horizontal plane. Initially OP=OQ, and the particles are projected with 

 equal velocities along the external bisectors of the angles OPQ, OQP respec- 

 tively. Prove that, throughout the motion, the tension of the cord varies 

 inversely as OP. 



79. Particles of masses M and m are attached to the ends of a thread, the 

 former being within a smooth fixed horizontal tube and the latter on a smooth 

 table in the horizontal plane of the tube. The thread is initially straight and 

 the particle of mass m is projected at right angles to the thread. Prove that 



the polar equation of its path is of the form r cos \6 A/J/-I | =c - 



80. Two particles, masses m, m', on a smooth horizontal table are 

 connected by a thread passing through a small smooth ring fixed in the table. 

 Initially the thread is just extended and in two straight pieces meeting at the 

 ring, the lengths of the pieces being a and a'. The particles are projected at 

 right angles to the string with velocities v and t/. Prove that, if T is the 

 tension at any time and r, / the distances from the ring, then 



r '3 ' 



Prove also that the other apsidal distances will be equal if 



