332 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



134. A cubical framework of twelve rods freely jointed at the corners is 

 suspended from a corner and held in shape by an elastic string occupying the 

 vertical diagonal. Prove that, if small oscillations take place with the string 

 remaining vertical, their period is the same as that of a simple pendulum of 

 length || (1-1 \ where I and 1 are the equilibrium length and natural length 

 of the string. 



135. A uniform rod rests in equilibrium on a rough gravitating uniform 

 sphere under no forces but the attraction of the sphere. Prove that, if 

 slightly displaced, it will oscillate in time 



where m is the mass of the sphere, a its radius, and 2,1 the length of 

 the rod. 



136. A uniform rod of length 2a moves in a smooth fixed tube under the 

 action of a fixed gravitating particle of mass m at a point distant c from the 

 tube. Prove that the period of small oscillations is 



137. A series of n infinitely long uniform circular cylinders, each of 

 radius c and mass M per unit of length, is ranged symmetrically round a 

 rigid framework freely moveable about a fixed axis J, the axis of each cylinder 

 being parallel to A and at distance a from it. They are attracted by a 

 similar gravitating fixed cylinder with a parallel axis at a distance &(>) 

 from A. Find the positions of stable equilibrium, and prove that the period 

 of small oscillation about such a position is 



where X is the pressure on the axis per unit of length, and the mass of the 

 framework is neglected. 



138. Two equal spheres, each of radius a and moment of inertia / about 

 an axis through its centre, have their centres connected by an elastic thread 

 passing through holes in their surfaces, and are set to vibrate symmetrically. 

 Prove (i) that, if in equilibrium the tension of the thread is T, then the time 

 of an oscillation of small amplitude a is 27ra~ 1 x /(7/7 T a), and (ii) that, if 

 the natural length of the thread is 2a and A is its modulus of elasticity, then 

 the period is 



4_ //2/\ /I dd 



2 V \\aJJ ov/(l- i sin 2 6}' 



139. Two equal uniform balls are fixed to the ends of a rod AB of 

 negligible mass which is suspended by its middle point by means of a wire 

 of such torsional elasticity that the system makes a complete oscillation 

 about in a horizontal plane in time T. Two equal fixed uniform spheres 

 of radius a are fixed with their centres at C, D so that AC and BD are each 

 of length b and are in the same horizontal plane with AB and perpendicular 

 to it on opposite sides. The attraction of the spheres alters the position of 



