EXAMPLES. 341 



188. A solid paraboloid of revolution is free to turn round its axis which 

 is vertical, and has a groove cut in its surface which makes a constant angle 

 a with the axis. A particle of mass m is placed in the groove at a depth k 

 below the vertex. Prove that when the particle has descended a depth h the 

 angular velocity of the paraboloid is 



/ 2gha {(h + A ) sin 2 a - a cos 2 a} 



\/ {I -\-4ma (A + A )} (/+4ma(a + A-fA- ) cos 2 a} 



where 4&amp;lt;x is the latus rectum of the paraboloid and / is its moment of inertia 

 about its axis. 



189. A uniform cube, of mass J/, and radius of gyration K about an axis 

 through its centre, rests on a smooth horizontal plane, and a smooth circular 

 groove of radius a is cut on the upper face and passes through the centre 

 of that face. A particle of mass m is projected along the groove from with 

 velocity F Prove that, if aty is the arc traversed by the particle, and &amp;lt; the 

 angle turned through by the block in any time, 



Hh 

 



where /3 2 = * 2 ( M + m)/4ma 2 . 



190. Two uniform rods each of length 2a are freely jointed and placed 

 on a smooth table in a straight line parallel to an edge. A cord is attached 

 to the joint and passing over the edge of the table at right angles supports a 

 body of mass l/n of that of either rod. Prove that the angle 6 through 

 which either rod has turned at time t is given by the equation 



(2 + n ( 1 + 3 sin 2 #)} ad 2 = 3g sin 6. 



191. Six equal uniform rods are freely jointed at a point and have 

 their other ends at the corners of a regular hexagon on a smooth horizontal 

 plane, these ends being connected by six similar elastic threads in the sides 

 of the hexagon. Initially all the threads have their natural lengths, and the 

 rods are inclined at an angle a to the vertical. Prove that the joint will or 

 will not reach the plane according as the ratio of the modulus of elasticity of 

 each thread to the weight of each rod &amp;lt; or &amp;gt; sin a cos a/(l - sin a) 2 . 



192. A rifled gun is mounted on a carriage without wheels. Prove that, 

 if a is the elevation of the gun, p the pitch of the barrel, k the radius of gyra 

 tion of the shot, and U, V the muzzle velocities of the shot when the carriage 

 is (1) fixed and (2) allowed a free recoil, then 



72 {2/^2 + sin 2 a + J/ COS 2 a /( Jf+ m )} = U 2 (1 + F/p 2 ) (sin 2 a + M 2 cos 2 a/( M + m) 2 }, 

 where m is the mass of the shot, and M is the mass of the gun and carriage. 



193. A particle is placed in a smooth elliptic tube of n times its mass at 

 an end of the major axis, and the tube is struck by a blow parallel to the 

 minor axis so that it starts to move parallel to this axis with velocity F. 

 Prove that the eccentric angle &amp;lt; of the position of the particle at any time 

 is given by the equation 



