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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.
 Winter '14
 Equations

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