280 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



equal, the particle P is always vertically under the middle point of AB and 

 thus its horizontal acceleration is \ (/+/ ) 



Hence (/+/ ) =f-f l sin a, 



giving /i sin aJ (/-/). 



Again, the horizontal acceleration of Q vanishes, and we have therefore 

 the acceleration / 2 of Q relative to B given by the equation 



/ 2 sin a =/ . 



Thus the accelerations of the particles are expressed in terms of / and / ; 

 in particular the vertical accelerations of P and Q are |(/-/ ) c ta and 

 / cot a downwards. 



Now let m be the mass of each particle and T lt T 2 , T 3 the tensions in 

 the threads as shown in the figure. Then resolving horizontally for A, P, 

 and B we have 



(T 2 -T l ) cos a, mf = (T 3 - T 2 ) cosa ...(1) ; 

 and resolving vertically for P and Q we have 



\m (/-/ ) cot a = - ( 2\ + T 2 ) sin a + mg, mf cot a = - 2T 3 sin a + mg. . .(2). 

 From the set of equations (1) we have 



TI cos a = mf, T 2 cos a = m (f /+ / ), T z cos a = m f (/+/ ) ; 

 and from (2), on substituting for T lt T 2 , T 3 , we have 



(/-/)coto + (5/+/)tano2 fl r, / cot a -f- 3 (/+/ ) tana =g; 



whence 



/ / 



&amp;gt; 

 4 cos 2a cos 2a 12 11 cos 2a + cos 2 2a&quot; 



248. Initial curvature. As an example of initial curvatures when the 

 motion does not start from rest we take the following problem : 



Two particles of masses m, m connected by an inextensible thread of length I 

 are placed on a smooth table with the thread straight, and are projected at right 

 angles to the thread in opposite senses. To find the initial curvatures of their 



Let u, v be the initial velocities of the particles, and o&amp;gt; the initial angular 



velocity of the thread, then 



\ ^ 



Let G be the centre of inertia 

 of the two particles. Then G 

 moves uniformly on the table 



with velocity 



(mu - m v)/(m + m ). 



y f 74^ The acceleration of G vanishes, 



and the acceleration of m relative 



to G is that of a particle describing a circle of radius m7/(m+m ) with angular 

 velocity &amp;lt;a; thus the acceleration of m along the thread is m la&amp;gt; 2 /(m + m }, and 



