# The motion of the carts can be described as a general

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Unformatted text preview: he procedure for ﬁnding the forces acting in such mechanical systems. 49 The easiest calculation of the forces is based upon the notion of work . On the one hand, the work required to pull a weight to a new position is equal to the increase in potential energy imparted to the weight. On the other hand, we have the equation Work = Force × Displacement, which implies that Force = Work Change in potential energy =− . Displacement Displacement Thus if V (x1 , x2 ) is the potential energy of the conﬁguration when the ﬁrst cart is located at the point x1 and the second cart is located at the point x2 , then the forces are given by the formulae F1 = − ∂V , ∂x1 F2 = − ∂V . ∂x2 In our case, the potential energy V is the sum of the potential energies stored in each of the three springs, V (x1 , x2 ) = 1 1 1 k1 x2 + k2 (x1 − x2 )2 + k3 x2 , 1 2 2 2 2 and hence we obtain the formulae claimed before: F1 = − ∂V = −k1 x1 + k2 (x2 − x1 ), ∂x1 F2 = − ∂V = k2 (x1 − x2 ) − k3...
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## This document was uploaded on 01/12/2014.

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