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Unformatted text preview: Cole Publishing Company,
2000. 94 instant of time. We will assume that the potential energy in the string when it
is in the conﬁguration u(x) is
L T
2 V (u(x)) =
0 2 du
dx dx, (4.17) where T is a constant, called the tension of the string.
Indeed, we could imagine that we have devised an experiment that measures
the potential energy in the string in various conﬁgurations, and has determined
that (4.17) does indeed represent the total potential energy in the string. On
the other hand, this expression for potential energy is quite plausible for the
following reason: We could imagine ﬁrst that the amount of energy in the string
should be proportional to the amount of stretching of the string, or in other
words, proportional to the length of the string. From vector calculus, we know
that the length of the curve u = u(x) is given by the formula
L 1 + (du/dx)2 dx. Length =
0 But when du/dx is small,
1
1+
2 du
dx 22 du
dx =1+ 2 + a small error, and hence
1 + (du/dx)2 1
is closely approximated by 1 + (du/dx)2 .
2 Thus to a ﬁrst order of approxim...
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 Winter '14
 Equations

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