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



99. A smooth horizontal circular wire rotates uniformly about a point in 

 its plane. Prove that the motion of a bead in the wire will be the same 

 as that of the bob of a simple pendulum. 



100. A particle is at rest in a smooth horizontal circular tube, and the 

 tube is made to rotate with uniform angular velocity about a vertical axis 

 through a point on the diameter passing through the particle. Prove that 

 the time of describing any arc bounded by a chord through the centre of 

 rotation is constant. 



101. A bead is initially at rest on a smooth circular wire of radius a in a 

 horizontal plane ; the wire is made to rotate with uniform angular velocity o&amp;gt; 

 about an axis perpendicular to its plane and passing through a point on the 

 diameter through the bead at a distance c from the centre. When the bead 

 has moved a distance ad on the wire, the wire is suddenly stopped. Prove 

 that the bead will subsequently move with velocity 



w (V( 2 + c 2 + 2ac cos 6)-(a + c cos 6}}. 



102. Two small beads of masses m lt m 2 slide along two smooth straight 

 rods which intersect at an angle a, and the beads are connected by an elastic 

 thread of natural length c and modulus X. The rods are made to revolve 

 uniformly in their plane, about their point of intersection, with angular 

 velocity &amp;lt;u. Prove that throughout the motion 



m 1 (r^ - r^w 2 ) + m 2 (r 2 2 - r 2 2 o&amp;gt; 2 ) + Xe 2 /c = const., 



where e is the extension of the thread, and r lt r 2 are the distances of the beads 

 from the intersection of the wires at any time. 



103. A smooth elliptic tube rotates about a vertical axis through its 

 centre perpendicular to its plane with uniform angular velocity &amp;lt;. Prove that 

 a particle can remain at an extremity of the axis major, and if slightly 

 disturbed will oscillate in a period 27r/v/(l e 2 )/eo&amp;gt;, where e is the eccentricity. 



104. A particle can move in a smooth elliptic tube which can turn about 

 its centre in a vertical plane, and, the major axis being vertical and the 

 particle being at rest at the highest point, the tube is suddeiily set in rotation 

 with uniform angular velocity *J{$gl(a + b)}, where 2a and 26 are the axes of 

 the ellipse. Prove that the particle will move on the ellipse as if under 

 a force to the centre varying as the distance. 



105. A body is describing an ellipse of semi-axes a, b about a centre of 

 gravitation, and when it is at a distance r from this centre it comes under the 

 influence of a small disturbing force directed to the same point and varying 

 inversely as the cube of the distance. Prove that the effect is the same as if 

 the body described under the original force an orbit which at the same time 

 rotated (with the body) round the centre of force with angular velocity n times 

 the angular velocity of the body, where n is a small constant such that the 

 semi-axes of this new free orbit are equal to those of the original one reduced 

 by fractions 2?i6 2 /r 2 and n (1 + 6 2 /r 2 ) of themselves. 



