o Show My Work opton polnts 2 p10 5.1 Your Teacher A model of a spring/mass syst

Question

o Show My Work opton polnts 2 p10 5.1 Your Teacher A model of a spring/mass system is 4x"+ e.x - o. By inspection of the differential equation only, discuss the behavior of the system over a long period of time. For large values of t the differential equation is approximated by x"-OThe solution of this equation is the linear funct on x- Thus, for large time, the restoring force will have Select. to the point where the spring is incapable of returning the mass, and the spring will smply o Show My Work (optienal Submit Answer Save Progresa Submilt Assignment Save Assignment Progress Home My Assignments 4 , 888t

Explanation / Answer

By the inspection of the differential equation we have :

4x''+ e-0.1tx = 0 and when considered over a long period of time

=> t tends to infinity

we have

e-0.1t = 0 as t tends to infinity ; [e-0.1t = 1/e0.1t = 1/ = 0] ;

hence we remain with the equation

4x'' = 0 which implies speed of the system remains constant after a long period of time as the system acceleration becomes zero ;

=> 4x' = c ----- where c is some constant ;

=> 4x = cx + d ; where d is another constant ;

=> we have

x(4 - c) = d

=> x = d/(4-c) ; for c not equal to 4 ;

=> we have the systems displacement becomes constant after a long period of time ;

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