MA 36600 LECTURE NOTES: MONDAY, APRIL 6ApplicationsSpring-Mass System:One Mass, Two Springs.Now say that we have a massmattached totwosprings with constantsk1andk2, respectively. We assume that these springs are attached to the oppositeends of a track of fixed length, with the massmin between. We continue to assume that the mass is underthe influence of an external forceF(t). Denotex=x(t) as thedisplacement from equilibriumon the trackat timet. We seek a differential equation forx.LetL1andL2denote the lengths to which the first and second springs are stretched at equilibrium,respectively. Using Hooke’s Law, we see that the force exerted on the mass from the first spring is−k1L1(pulling to the left), whereas the force exerted on the mass from the second spring is +k2L2(pulling to theright). Since we are at equilibrium, the sum of the forces must be zero:−k1L1+k2L2= 0L1+L2==⇒L1=k2k1+k2,L2=k1k1+k2.
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