838 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



particle is slightly disturbed, the length of the simple equivalent pendulum 

 for the small oscillations is 



rp cos aj(r + 3p cos 2 a sin a), 



where p is the radius of curvature of the meridian curve and a the inclination 

 of the normal to the vertical at any point on the horizontal circle. 



171. A thread of length I has its ends attached to two points distant c 

 apart on a vertical axis, and a bead can slide on the thread ; the system 

 rotates about the vertical axis with angular velocity <. Prove that, if 



the time of a small oscillation about a position of relative equilibrium is 



where A = 2yP/ a (I 2 - c 2 ). 



172. A particle describes a circle uniformly under the influence of two 

 centres of force which attract inversely as the square of the distance. Prove 

 that the motion is stable if 3 cos 6 cos <<!, where 6, <f> are the angles which 

 a radius of the circle subtends at the centres of force. 



173. A straight uniform rod passes through a ring on a smooth horizontal 

 plane, and an elastic thread whose natural length is equal to that of the rod 

 has its ends fastened to the ends of the rod and its middle point fixed to the 

 ring. Prove that, if the rod is rotating about its centre with an angular 

 velocity such that the steady motion is unstable, then if it is slightly dis- 

 turbed its centre will describe the curve whose polar equation is 



(I + K 2 /r 2 ) sin 2 a = cosh 2 (6 sin a), 



where 6 is measured from the apse line, K is the radius of gyration of the rod 

 about its centre, and K tan a is the value of r at the apse. 



174. A uniform rod of length 26 can slide with its ends on a smooth 

 vertical circular wire of radius a and the wire is made to rotate about a 

 vertical diameter with uniform angular velocity co. Prove that the lowest 

 horizontal position is stable if 



175. Four equal uniform rods are freely jointed so as to form a rhombus 

 ABDC ; AB, AC are connected with a vertical spindle Ly means of a hinge 

 at A, permitting free motion in the vertical plane BAG. An elastic thread, 

 of natural length %AD when AB is inclined to the vertical at an angle a, and 

 of modulus of elasticity equal to twice the weight of a rod, joins A to D. 

 Prove that if the system is started to rotate with angular velocity CD, 



when each rod makes an angle a with the vertical the system will move 

 steadily, and that the time of a small oscillation about the steady motion is 



(7r/o))V(l+3sin 2 a). 



