271-273] MOTION OF A CHAIN. 309 



This equation serves to determine the initial tension at any 

 point of the chain. To determine the arbitrary constants which 

 enter into the solution of the equation we have to use the condi- 

 tions which hold at the ends, or at other special points, of the 

 chain. Thus if one end of the chain is guided to move on a given 

 curve the acceleration of the extreme particle must be directed 

 along the tangent to the curve. Cases arise in which this method 

 cannot be applied ; thus, in the case of a heavy chain with an end 

 moveable on a smooth straight wire not perpendicular to the 

 tangent at the end, the equation of motion of an element at the 

 end, found by resolving along the wire, cannot be satisfied if the 

 acceleration of the element is finite (not infinite) and the tension 

 is finite (not zero). The conclusion in such cases must be that 

 the chain becomes slack at the end, and it may become slack 

 throughout. In such cases it is usually convenient to suppose the 

 end of the chain attached to a ring which can slide on the wire, 

 and to take the mass of the ring, at first, to be finite : when the 

 problem has been solved with this condition we can pass to the 

 case above described by supposing the mass of the ring to be 

 diminished without limit. 



*273. Impulsive Motion. The equations of impulsive mo- 

 tion when the chain is suddenly set in motion follow at once by 

 the method of Article 269. We have only to regard 8 and N as 

 the resolved parts of an impulse reckoned per unit of mass applied 

 to an element, and T as impulsive tension. The equations are 



dT 



mu = = + mS y 



OS 



T 



rnv = \- mN. 



P 



The kinematic conditions are the same as for a chain in con- 

 tinuous motion, viz.: equations (1) and (2) of Article 269. 



In case no impulses are applied to the chain except at its ends, 

 S and N vanish, and we can eliminate u and v, obtaining an 

 equation for T in the form 



a 



as \m osj mp- 



The solution of this equation subject to the given terminal 

 conditions gives the impulsive tension at any point of the chain. 



