358 



BELL SYSTEM TECHNICAL JOURNAL 



where x^ is the displacement of m^ measured downward from its position at 

 first contact of the spring with the floor (see Fig. 1.2.1). For a non-linear 

 spring P will be some other function of x^ : 



P = P{x->). 



(1.2.2) 



To write the equation of motion for the mass mi , we consider the forces 

 acting on it at any instant. These are (see Fig. 1.2.2(b)) the spring force 

 P and the weight ntig, where g is the acceleration of gravity. When Xi 

 is positive (i.e., a downward displacement of m2 from its position at first 

 contact of the spring with the floor) the spring exerts an upward force P 



/ 



floor 



/ / /'/ y / / y / / // // /^/ //y' 



(a) (b) 



Fig. 1.2.1 — Elementary system. 



(c) 



(a) (b) 



Fig. 1.2.2 — Free body diagram for elementary system. 



(a) Spring not in contact with floor. 



(b) Spring in contact with floor. 



on the mass, opposing the weight. The total downward force on m-i is 

 thus mig — P. By the second law of motion, the product of the mass and 

 its acceleration at any instant is equal to the appUed force : 



mix% = mig — P, 



(1.2.3) 



where the symbol X2 , representing the acceleration of W2 , stands for the 

 second derivative of displacement with respect to time {d'xi/df). Equation 

 (1.2.3) is the law governing the motion of W2 as long as the spring is in con- 

 tact with the floor. When the spring is not in contact with the floor, it can 

 exert no force on the mass so that, in writing the equation of motion that 



