EXAMPLES. 347 



226. An elastic string (modulus X, mass ma, unstretched length a,) is 

 confined within a straight tube to one end of which it is fastened, and the 

 tube rotates about that end with uniform angular velocity o> in a horizontal 

 plane. Show that the length of the string in equilibrium is 



tan 6 . Im 



a , where 6 = aa> . /y- 



227. A smooth circular tube of radius a having a narrow slit on its 

 inner side rotates about a fixed vertical diameter AB with angular velocity o>, 

 A being the highest point, and a uniform chain which subtends an angle 2a 

 at the centre can move in the tube. To its upper end is attached an elastic 

 thread which passes through the slit and through a ring at A and is fastened 

 at B, its natural length is 2a and its modulus of elasticity equal to a weight 

 4a tan a of the chain. Prove that the chain is at rest in the tube in the 

 same position for all values of o>, and that, if slightly disturbed, it oscillates 

 in a period 



27iV{aa cosec aj(g tan a + aa> 2 cos a)} . 



228. A cone of vertical angle 2a, whose moment of inertia about its axis 

 is /, is free to turn about its axis which is vertical, and a fine smooth groove 

 is cut on its surface so as to make a constant angle /3 with the generators. 

 A uniform chain of mass /z and length I moves in the groove under gravity, 

 one end being initially at the vertex. Prove that, if 6 is the angle through 

 which the cone has turned when the upper end is at a distance r from the 

 vertex, 



{7cosec 2 a//i+i2 cos 2 }e 2 * sinacot 0=r 2 + r cos/3 + ^ 2 cos 2 0+7 cosec 2 a/p. 



229. A uniform chain of mass m and length 2 is in a tube of uniform 

 bore in the form of an equiangular spiral which revolves in its plane about 

 its pole with uniform angular velocity o>. Prove that the tension at any 

 point of the chain is \m cos 2 a (I 2 # 2 ) a> 2 /, where a is the angle of the 

 spiral and x the arcual distance of the point from the middle point of 

 the chain. 



230. A smooth tube in the form of a cycloid generated by a circle of 

 radius a rotates uniformly about the base of the cycloid with angular velocity 

 G, and a piece of uniform chain of length 21 is in the tube. Prove that, if 

 the chain is under no forces but the pressure of the tube, the time of a small 

 oscillation about the position of relative equilibrium is 



231. A rough circular cylinder of radius c is fixed with its axis vertical, 

 and a uniform chain lying on a smooth horizontal plane has a length c/3 in 

 contact with the cylinder, its end portions of lengths a and b being straight. 

 The free end of the length a is pulled by a constant force F in the direction 

 of its length. Prove that when the free end of the length b reaches the 

 cylinder it will be moving with a velocity 



/(IF l*-(l-b? ] 

 V l2--4j' 



