278 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



6. A rectangle formed of four uniform rods, of lengths 2a and 26 and 

 masses m and m , freely hinged together, is rotating in its plane about its 

 centre with angular velocity n when a point in one of the sides of length 2a 

 becomes suddenly fixed. Prove that the angular velocity of the sides of 

 length 26 instantly becomes %n(3m + m )/(3m + 2m ), and find the angular 

 velocity of the sides of length 2a. 



INITIAL MOTIONS. 



245. Nature of the problems. We suppose that a system 

 is held in some definite position in a field of force, and that at a 

 particular instant some one of the constraints ceases to be applied, 

 then the system begins to move, each particle of it with a certain 

 acceleration. Our first object in such a case is to determine the 

 accelerations with which the parts of the system begin to move. 

 When the accelerations have been found there is generally no 

 difficulty in determining the initial values of the reactions of 

 supports, or internal actions between different bodies of the system ; 

 and the determination of the unknown reactions is our second 

 object. 



Again, we may enquire as to approximate expressions for the 

 coordinates of a particle in terms of the time elapsed since the 

 instant when the constraint was removed. For motions starting 

 from rest, a knowledge of the accelerations alone determines the 

 initial tangents to the paths of the particles, and gives values for 

 the displacements of the particles after a short time which are 

 correct to the second order of small quantities, the time being of 

 the first order. If the approximation could be continued beyond 

 the second order we could state the curvature of the path of each 

 of the particles. Thus a convenient way of referring to problems 

 of this character is to regard as their object the determination of 

 initial curvatures. 



Finally we may remark that the problem of determining the 

 curvature of the path of a particle whose velocity is not zero 

 offers no difficulty when the velocity and acceleration are known, 

 since the resolved acceleration along the normal to the path is 

 the product of the square of the resultant velocity and the 

 curvature. This remark enables us easily to determine the initial 

 curvature of the path of a particle when its motion is changed 

 impulsively. 



