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Unformatted text preview: rium position of a system of springs. We consider a collection of n masses in R2 , with
locations (x1 , y1 ), . . . , (xn , yn ), and masses m1 , . . . , mn . (In other words, the vector x ∈ Rn gives
the xcoordinates, and y ∈ Rn gives the ycoordinates, of the points.) The masses mi are, of course,
positive.
For i = 1, . . . , n − 1, mass i is connected to mass i + 1 by a spring. The potential energy in the ith
spring is a function of the (Euclidean) distance di = (xi , yi ) − (xi+1 , yi+1 ) 2 between the ith and
(i + 1)st masses, given by
0
di < li
Ei =
2 d ≥l
(ki /2)(di − li )
i
i
where li ≥ 0 is the rest length, and ki > 0 is the stiﬀness, of the ith spring. The gravitational
potential energy of the ith mass is gmi yi , where g is a positive constant. The total potential energy
of the system is therefore
n −1 Ei + gmT y. E=
i=1 The locations of the ﬁrst and last mass are ﬁxed. The equilibrium location of the other masses is
the one that minimizes E .
(a) Show how to ﬁnd the equilibrium pos...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.
 Fall '13
 F.Borrelli
 The Aeneid

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