EXAMPLES. 331 



128. A uniform solid right circular cone of height h, vertical angle 2a, 

 and radius of gyration k about an axis through its centre of inertia at right 

 angles to its axis of figure, rests with its vertex downwards between two 

 rough parallel rails at a distance 2c apart in a horizontal plane. Prove that, 

 if the equilibrium is stable, the period of the small oscillations about it is 



7r N /[{16& 2 sin 2 a + (3 sin a - 4c cos af\\g sin a cos a (4c- 3h tan a)]. 



129. A uniform sphere of radius c is placed on a horizontal wire in the 

 form of an ellipse of axes 2a, 26. Prove that, if the wire is rough enough to 

 prevent slipping, the length of the simple equivalent pendulum of the small 

 oscillation about the position of equilibrium is 



where F=|c 2 , and c^ = c 2 -6 2 . 



130. Two equal wheels each of mass J/, radius a, and radius of gyration 

 k about its axis, are rigidly connected by an axle of length c and run on a 

 horizontal plane. Two particles, each of mass m, are connected, one to each 

 of the centres of the wheels by cords which pass over smooth pegs in the 

 line of centres. Prove that if the wheels are symmetrically placed between 

 the pegs, and slightly displaced by rolling on the plane, the time of a small 

 oscillation is 



where 26 + c is the distance between the pegs. 



131. A solid circular cylinder, bounded by two planes making given 

 angles with the axis, is laid on its curved surface on a rough horizontal plane. 

 Find the position of stable equilibrium, and prove that, if I is the length of 

 the simple equivalent pendulum for a small oscillation, and d the diameter of 

 the cylinder, then the ratio of the longest and shortest generators is 



132. Four equal rods each of length 2a are jointed so as to form a 

 rhombus which is set up in a vertical plane with its lowest corner fixed and 

 one diagonal vertical, being kept in shape by an elastic thread in the other 

 diagonal, and in the position of equilibrium the thread is stretched to twice 

 its natural length and the rods make equal angles a with the vertical. Prove 

 that the period of a small symmetrical oscillation is the same as that of a 

 pendulum of length 



a sec 2a cos a (1+ 3 sin 2 a). 



133. Four equal uniform rods, each of length 2a and weight JT, are 

 freely jointed so as to form a rhombus, and the opposite corners are joined by 

 two similar elastic threads of equal unstretched lengths and of modulus A. 

 Prove that, if the system is laid on a smooth horizontal plane and the threads 

 never become slack, each rod swings about its position of equilibrium like a 

 simple pendulum of length | v ^2 U'a/A. 



