324 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



86. Three particles A, JB, C are connected by two threads Afi, AC and 

 placed in a line on a smooth table. The extreme particles are projected at 

 right angles to the thread with velocities u, v. Prove that, if m, p, q are the 

 masses, and a, b the lengths of the threads, the initial curvatures of the 

 paths of B and C are 



a + qv 2 /b , (p+m)v 2 /b+pu 2 /a 



87. A particle of mass m is attached to one end of a thread which passes 

 through a bead of mass M and the other end is secured to a point on a 

 smooth horizontal table on which whole rests. Initially the two portions of 

 the thread are straight and contain an obtuse angle a, the portion between m 

 and M being of length a, and m is projected at right angles to this portion. 

 Prove that the initial radius of curvature of the path of m is 



88. Into the top of a smooth fixed sphere of radius a is fitted a smooth 

 vertical rod. A uniform rod of length 26 rests on the sphere with its upper 

 end constrained to remain on the vertical rod, the centre of gravity being at 

 a distance c from the point of contact. Prove that if the constraint is 

 removed the pressure on the sphere is instantaneously diminished in the ratio 



6(6-c) : (& 2 + 3c 2 ). 



89. Two equal uniform rods each of length 2a are freely hinged at one 

 extremity and their other extremities are connected by an inextensible thread 

 of length 2. The system rests on two smooth pegs distant c apart in a 

 horizontal line. Prove that, if the thread is severed, the initial angular 

 acceleration of either rod is 



(8a 2 c - 1 3 ) gl(l a?l 2 + 32a 4 c 2 /Z 2 - 8a?cl). 



90. Six equal uniform rods each of mass m are freely jointed and are 

 kept in the form of a regular hexagon by a thread joining two opposite 

 corners. The hexagon is in a vertical plane with one rod- fixed and horizontal 

 and a particle of mass p is attached to the middle point of the lowest rod. 

 The thread is then destroyed. Prove that p descends with initial acceleration 



91. A uniform circular disc is supported in a vertical plane by two cords 

 attached to the ends of a horizontal diameter, each making an angle a with 

 the horizontal. One of the cords being cut, prove that the tension of the 

 other is suddenly diminished in the ratio 2 sin 2 a : 1 + 2 sin 2 a. 



92. A uniform circular disc is symmetrically suspended by two elastic 

 cords of natural length c inclined at an angle a to the vertical and attached 

 to the highest point of the disc. Prove that, if one of the cords is cut, the 

 initial radius of curvature of the path of the centre of the disc is 



3 cos a b (b - c)/(c sin 4a - b sin 2a), 

 where b is the equilibrium length of each cord. 



