286 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



*252. Examples. 



1. A uniform rod of length 2a and mass m is symmetrically supported 

 in a horizontal position by two inextensible cords attached to its ends each of 

 length I and making an angle a with the vertical. Prove that if one cord 

 is cut the tension in the other immediately becomes w^cosa/(l + 3cos 2 a) 

 and that the ratio of the initial angular accelerations of the cord and rod is 

 ^ (a/I) tan a sec a. 



2. Two equal uniform rods are freely jointed at common ends, the other 

 end of the first is fixed so that the rods can turn about it, and the other end 

 of the second is held at the same level as the fixed end of the first, so that 

 the rods make equal angles a with the horizontal, and this end is let go. 

 Prove that the initial angular accelerations of the rods are in the ratio 



6-3 cos 2a : 9 cos 2a- 8. 



3. A uniform triangular disc is supported horizontally by three equal 

 vertical cords attached to its corners. Prove that, if one cord is cut, the 

 tension of each of the others is instantly halved. 



4. Three equal uniform rods are freely jointed at B and C so as to form 

 a quadrilateral A BCD, and the ends A and D can slide on a smooth horizontal 

 rod. The system is initially held (by means of horizontal forces applied at 

 A and D} in a symmetrical position with BC lowest and horizontal, and with 

 AB and CD equally inclined at an angle a to the horizontal. Prove that, 

 when the ends A and D are released, the pressures at A and D are changed 

 in the ratio 1 -f sin 2 a : 5 3 sin 2 a. 



5. A uniform rod of length 2a is placed at an inclination a to the hori 

 zontal in contact with a smooth peg at its middle point. Prove that the 

 initial radius of curvature of the path of a particle distant r from the middle 

 point is ( 2 /r) tan a. 



6. Two equal uniform rods A B, BC each of length a are freely jointed 

 at B and can turn freely about A. Prove that, if the system is released from 

 a horizontal position the initial radius of curvature of the path of C is fa. 



SMALL OSCILLATIONS. 



253. We have to consider the small motion of a system 

 which is slightly displaced from a position of equilibrium. We 

 confine our attention to cases where any position of the system is 

 determined by assigning the value of a single geometrical quan 

 tity 0, as in the case of the simple circular pendulum (Article 189). 

 We can always choose to vanish in the position of equilibrium ; 

 for, if it has been chosen in any other way so that its value in the 

 position of equilibrium is , then 6 can be used instead of 6. 



