3o8 THE GRAMMAR OF SCIENCE 



and inextensible string. Such a string would not in 

 itself alone produce sensible accelerations in A or B. Since 

 the string is inextensible, the whole system must move in 

 the same direction, say from right to left. Then clearly 

 the velocity of A must be at all times equal to the 

 velocity of B, or the string would be stretched. But if 

 the velocities of A and B are always equal, their accelera- 

 tions must also be equal, or their velocities, being differ- 

 ently spurted, would begin to differ. Hence we conclude 

 that the total acceleration of A towards the left must be 

 equal to the total acceleration of B in the same direction, 

 or in symbols : — 



g-fba=fah-g ■ ■ ■ ■ (i-)- 



But by the fifth law of motion {i.e. (7), p. 303) 



fha Wb r- \ 



-^ = .... (11.. 



fab '"a 



Thus (i.) and (ii.) are two simple relations to find /^^ and 

 y^i. By elementary algebra we have : — 



/««' = ^ .„ , .„ S ^ and f^a = 2- 



Hence we deduce : — 



Acceleration of A or B to the left = 4'-- //,« = -"—^o- (iii.). 



ma + Jnf 



Further : — 



Force of B on A = mass of A x acceleration of A due to B, 



= W„X_/to, 



= 2 -p- 



nta + mf 



= in}^y.Jabi or Force of A on B. 



Now this force of B on A is what we usually term 

 the tension in the string. Hence we have : — 



Tension m the string = 2 g . . (iv.). 



A further important point has now to be noticed. In 

 order that A and B should be at rest relative to the 

 field which produces the acceleration g, it will be neces- 

 sary that their velocities should always be zero, and this 

 involves that the changes in their velocities, or their 



