ILLUSTRATIVE EXAMPLES 



203 



The tension of the string must be the same throughout, say T. If the accel- 

 erations of the pulleys are /p, /Q, , all measured down, we have equations 

 of motion of the type 



Pg-2T=Pf P , 



(a) 



one equation for each pulley. The unknown 

 quantity T enters these equations, as well as 

 the n unknown quantities /p, /Q, . Thus 

 there are n + 1 unknown quantities, and so 

 far only n equations connecting them. An- 

 other equation is therefore required, and this 

 is obtained by noticing that the accelerations 

 /p, /Q, cannot be independent, for the 

 length of the string must remain unaltered. 



Let us denote the depths of P, Q, below the horizontal ring by SP, 



Then S P + S Q -\ ---- 



must be constant throughout the motion. It follows that 



/P+/Q+ =(). 



Substituting the values of /p, 



from equation (a), we obtain 



so that 



ng 



-4- | 

 i^ 



and on substituting this value for 2 T in equation (a), we obtain the required 

 value of /p. 



EXAMPLES 



1. Show that the tension of the string in an Atwood's machine is intermediate 

 between the weights of the two masses. Show also that it is nearer to the 

 smaller than to the larger of these weights. 



2. Two weights 16 and 14 ounces respectively are connected by a light inex- 

 tensible string which passes over a smooth pulley. The weights hang with the 

 strings vertical and the string is clamped so that no motion can take place. If 

 the string is suddenly undamped find the change in the pressure exerted on the 

 pulley. 



3. A string passing across a smooth table at right angles to two opposite 

 edges has attached to it at the ends two masses P, Q which hang vertically. 

 Prove that, if a mass M be attached to the portion of the string which is on the 

 table, the acceleration of the system when left to itself will be 



P-Q 



