EXAMPLES. 211 



position in which the threads and the radii through A and B form a square 

 with C vertically below the centre. Prove that when A and B meet the 

 velocity of either of them is 



69. Two particles of masses M, m are connected by a cord passing over a 

 smooth pulley, the smaller (m) hangs vertically and the other (M) moves in a 

 smooth circular groove on a fixed plane of inclination a to the vertical, the 

 highest point of the groove being vertically under the pulley. M starts from 

 the highest point of the groove without initial velocity. Prove that, if it 

 makes complete revolutions, the radius of the groove must not exceed 



hmMcos a/(m 2 - M 2 cos 2 a), 

 where h is the height of the pulley above the highest point of the groove. 



70. A particle moves from rest at an extremity of the major axis of a 

 smooth elliptic groove of axes 2a, 26 cut in a horizontal table, being attached 

 to a thread which passes through a small hole at the centre of the ellipse and 

 supports a particle of equal mass. Prove that the horizontal pressure on the 

 groove when the first particle is at an extremity of the minor axis vanishes if 



71. A particle of weight W moves in a smooth elliptic groove on a 

 horizontal table, and is attached to two threads which pass through holes at 

 the foci, and each thread supports a body of weight W. One of the bodies is 

 pulled downwards with velocity Ve when the particle is at an end of the minor 

 axis. Prove that, if F 2 <a6 2 ^/{e(3a 2 -26 2 )}, the threads do not become slack, 

 and that in this case the horizontal pressures, R and R', on the groove when 

 the particle is at the ends of the axes are connected by the equation 



Rb 3 ~ R'a (3a 2 - 26 2 ) = 6 Wa?be 2 , 

 where 2a and 26 are the principal axes, and e is the eccentricity of the ellipse. 



72. A smooth parabolic wire, on which is a smooth bead of weight w, is 

 fixed in a horizontal plane. To the bead is attached a thread, which passes 

 through a smooth ring fixed at the focus of the parabola and carries, at its 

 other end, a weight wl(e\}. Prove that the tension T of the thread at any 

 stage of the motion is given by an equation of the form 



(eT- w} (er - a) 2 = const., 



where r is the focal distance of the bead and 4a the latus rectum of the 

 parabola. 



73. Two smooth straight horizontal non-intersecting wires are fixed at 

 right angles to each other at a distance d apart. Two small rings of equal 

 mass, connected by an inextensible thread of length I, slide on the wires, and 

 they are projected with velocities u and v from points at distances a and b 

 from the shortest distance between the wires. Prove that after the thread 

 becomes tight the motion is oscillatory and of period 2rr (I 2 - d?}l(av ~ bu). 



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