342 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



where 2a and 26 are the principal axes of the tube, and k is its radius of 

 gyration about its centre. 



194. Two rough horizontal cylinders each of radius c are fixed with their 

 axes inclined to each other at an angle 2a and a uniform sphere of radius a 

 rolls between them, starting with its centre very nearly above the point of 

 intersection of the highest generators. Prove that the vertical velocity of its 

 centre in a position in which the radii to the two points of contact make angles 

 <f> with the horizontal is 



J{lOg (a + c) (1 - sin <)/(7 - 5 cos- 2 a cos 2 <)}. 



195. Two equal right circular cones of semi vertical angle a are fixed with 

 their axes horizontal so as to touch along a horizontal generator and to have 

 their vertices coincident. A sphere of radius a rolls between them. Prove 

 that the height z of its centre above the plane of the axes satisfies the 

 equation 



( 2 2 F / z 2 \ 2 } 



^ 2 j 1 + a* C t2 " + & C t2 a V e 2 " + lfi) J =ff ^ ~ *)' 



196. A circular tube of mass m and radius a contains a particle of mass 

 win, and the tube rotates freely about a vertical chord AB (A above B] which 

 subtends an angle 2a at the centre. Initially the particle is at the highest 

 point C of the tube and the tube is set rotating with angular velocity G. 

 Prove that, if the particle oscillates between C and B, then 



aQ 2 {(7i+l)cos 2 a+} cos 2 a =g(l+ sin a) (1 + 2 cos 2 a). 



197. A rhombus of four equal uniform rods each of length a freely 

 jointed together is laid on a smooth horizontal table with one angle equal to 

 2a, and the opposite corners are joined by similar elastic threads of natural 

 lengths 2a cos a and 2a sin a. Prove that, if one thread is slightly extended 

 and the rhombus left free, the periods during which the threads are extended 



in the subsequent motion are in the ratio (cos a)* : (sin a)*. 



198. A particle of mass m is placed in a smooth straight tube which can 

 rotate in a vertical plane about its middle point, and the system starts from 

 rest with the tube horizontal. Prove that the angle 6 which the tube makes 

 with the vertical when its angular velocity is a maximum and equal to o> is 

 given by the equation 4 (mr 2 + 1} a>* - 8mgra> 2 cos + w<7 2 sm 2 0=0, where / 

 is the moment of inertia of the tube about its middle point, and r is the 

 distance of the particle from that point. 



199. Four equal rods of length a and mass m are freely jointed so as to 

 form a rhombus one of whose diagonals is vertical ; the ends of the horizontal 

 diagonal are joined by an elastic thread at its natural length, and the system 

 falls through a height h to a horizontal plane. Prove that, if any rod makes 

 an angle 6 with the vertical at time t after the impact, then 



a 2 1+3 sin 2 a a ^ 2ma sin a 



where a is the initial value of #, and X is the modulus of the thread. 



