244-247] 



INITIAL MOTIONS. 



279 



246. Method for initial accelerations. It is always 

 possible to determine expressions for the accelerations of all the 

 points of a connected system in terms of a small number of 

 independent accelerations, and there is always the same number 

 of equations of motion free from unknown reactions, so that all 

 the accelerations can be found. The expression of the initial 

 accelerations in the manner proposed is facilitated by observing 

 (1) that the velocity of every particle initially vanishes, (2) that 

 every composition and resolution may be effected by taking the 

 position of the system to be that from which it starts. The 

 method laid down will be better understood after the study of an 

 example. We purposely choose one of a somewhat complicated 

 character in order to illustrate the points mentioned. 



247. Illustrative Problem. Four equal rings A, B, (7, D are at equal 

 distances on a smooth fixed horizontal rod, and three other equal and similar 

 rings P, Q, R are attached by pairs of equal inextensible threads to the pairs of 

 rings (A, B), (B, (7), (C, D). The system is held so that all the threads initially 

 make the same angle a with the horizontal, and is let go. It is required to find 

 the acceleration of each ring. 



From the symmetry of the system the accelerations of A t D are equal and 



Fig. 73. 



opposite, so are those of B, C, and those of P, R. Also the acceleration of Q 

 is vertical. 



Let/, /' be the accelerations of A, B along the smooth horizontal rod. 



Now relative to A, P describes a circle, and thus the acceleration of P 

 relative to A is made up of a tangential acceleration/! at right angles to AP, 

 and a normal acceleration proportional to the square of the angular velocity 

 of A P. Since the initial angular velocity vanishes, we have, as the relative 

 acceleration,/! at right angles to AP. Again, since the threads AP, BP are 



