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~A sin a, 



giving /! sin a = (/-/). 



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 f ; 

 in particular the vertical accelerations of P and Q are ^ (/-/') cot a and 

 /' cot a downwards. 



Now let m be the mass of each particle and T ly T^ T 3 the tensions in 

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

 and B we have 



m/^coso, im (/+/') = (^ 2 -^) cos a, mf = (T 3 - T 2 ) cosa ...(1); 

 and resolving vertically for P and Q we have 



\m (/-/') cot a = - ( TI + T 2 ) sin a + mg, mf cot a = - 2 T s sin a 4- mg. . . (2 ). 

 From the set of equations (1) we have 



r i cosa = m/, ^ cos a = m (f/+/), T 3 cos a = mf (/+/'); 

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



(/-/') cot a + (5/+/) tana = 2#, / cot a +3 (/+/') tan a=g-, 



. f f _ #sin2a 



~ "12-11 cos 2a + COS 2 2a' 



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 

 paths. 



Let u, v be the initial velocities of the particles, and o> the initial angular 



velocity of the thread, then 



u+v=la>. 

 Let G be the centre of inertia 



in/ 



of the two particles. Then 



moves uniformly on the table 

 with velocity 



(mu m'v)/(m + m'}. 



j t 74. The acceleration of G vanishes, 



and the acceleration of m relative 



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

 velocity o>; thus the acceleration of m along the thread is m'la> 2 /(m + m'), and 



