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/ /s /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 . / -7 . 



181 \/ 3 



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

 have initial angular accelerations a> 1) &> 2 , ... a n in one plane. Prove that, if 

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



(!! + a 2 o> 2 + . . . 4- a n a> n }*l(a l0 >* + 2 a> 2 2 + . . . + n o> n 2 ). 



111. A system of two equal uniform rods AS, 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 A BCD 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 m lt m 2 , ra 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 



2 



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 m, M, m are attached to the points B, C, 

 D of B, thread AE of length 4a, and rest suspended by the ends A, JS'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/"-f-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 



V 11 





