328 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



109. Two rods AC, CB of equal length 2a are freely jointed at C, the rod 

 AC being free to turn in a vertical plane about the point A, and the end B 

 of the rod CB being attached to A by an inextensible string of length 4a/ N /3. 

 The system being in equilibrium the string is cut. Show that the initial 



4 /41 3 

 radius of curvature of the path of B is a -^ ./ . 



110. A set of n equal rods are jointed together in one straight line and 

 have initial angular accelerations &amp;lt;o 1 , co 2 , ... &amp;lt;o n in one plane. Prove that, if 

 one end is fixed, the initial radius of curvature of the path of the free end is 



111. A system of two equal uniform rods AJ3, CD and a sphere of 

 diameter BC equal to the length of either rod is free to turn about A, the 

 bodies being freely jointed at B and C, and ABCD being initially a horizontal 

 straight line. Prove that, if the mass of the sphere is equal to that of either 

 rod, the initial radius of curvature of the path of D is \ 



112. Three particles, of masses MJ, i 2 , w 3 , are symmetrically attached 

 to a circular wire of negligible mass and of radius a which can move in a 

 smooth circular tube of the same radius fixed in a vertical plane. Prove 

 that the length of the simple equivalent pendulum of the small oscillations 

 of the system is 



113. Two equal particles of mass Psina are attached, at distance 

 2a sin a apart, to a thread, to the ends of which particles of mass P are 

 attached. The thread is hung over two pegs distant 2a apart in a horizontal 

 line. Prove that the period of the small oscillations about the position of 

 equilibrium is the same as that for a simple pendulum of length a tan a. 



114. Three particles of masses wi, M, m are attached to the points B, C, 

 D of a thread AE of length 4a, and rest suspended by the ends A, ^from 

 two points at the same level. The portions AB, BC, CD, DE are each of 

 length a and make with the horizontal angles a, /3, /3, a respectively. Prove 

 that J/tana = (J/+2m)tan/3, and that, if M receives a small vertical dis 

 placement, the period of the small oscillations is the same as for a simple 

 pendulum of length 



sin a sin /3 sin (a - /3) cos (a - /3) 

 sin 2 a cos a -f- sin 2 /3 cos /3 



115. A particle of mass M is placed near the centre of a smooth circular 

 horizontal table of radius a ; cords are attached to the particle and pass over 

 n smooth pulleys placed symmetrically round the circumference, and each 

 cord supports a mass M. Show that the time of a small oscillation of the 

 system is 



