202 MOTION UNDEK CONSTANT FORCES 



If, however, the system is known to have been started in motion in the direc- 

 tion assumed, then the acceleration given by equation (c) will be in operation, 

 increasing the velocity if positive, and decreasing it if negative. In the latter 

 case the system will in time be reduced to rest, and we must then examine 

 whether or not it will start into motion in the reverse direction. 



2. To one end of the string of an Atwood's machine a weight of mass mi is 

 attached. To the other end a smooth pulley of mass m% is attached, over which 

 passes a string with masses m^, m hanging at its ends. Find the motion. 



Let the mass mi be supposed to have an acceleration /, measured downwards. 

 Then m 2 must have an acceleration / upwards. The masses m 3 , m 4 will them- 

 selves form an Atwood's machine, the whole of which 

 moves upwards with an acceleration/. Thus the ten- 

 sion in the string of this machine, say TI, is (cf . 163) 



f 



(a) 



m 2 If we denote the tension in the string connecting mi 

 and m 2 by T 2 , we have as the equation of motion of m 2 , 



while the equation of motion of mi is 



FIG. 114 



Eliminating 2\ and T 2 from equations (a), (6), and (c), we obtain as the 

 value of the acceleration /, 



mi m 2 



f 



The accelerations of the masses 



, m* relative to m 2 are known, by 163, 



3. At equal intervals on a horizontal circle n small smooth rings are fixed, 

 and an endless string passes through them in order. If the loops of the string 

 between each consecutive pair of rings support n pulleys of masses P, Q, R, 

 respectively, the portions of string not in contact with the pulleys being vertical, 

 show that the pulley P will descend with acceleration 



