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Unformatted text preview: Physics 316, Problem Set 3, due Tuesday October 18 in class 1. method of similarity: ( cf. Chapter 2 Section 11) The text demonstrates a way of inferring dynamical behavior by analyzing the scaling properties of the equation without solving it. This problem is meant to show these procedures. Here we wish to find the limiting properties of a systems motion as the energy of a particle goes to zero. The potential energy function U ( x ) of a point particle has the form of two semicircles side by side: U ( x ) = U p 1- (1-| x | ) 2 . A particle with energy E U is trapped in the region near x = 0. It is obliged to oscillate with some period . The object is to find how the period varies as the energy E 0 by the method of similarity. That is, define scaled variables E E , t t and x x . Then determine how and must vary with in order to leave the equation of motion invariant. a) How do the maximum displacement in x and the temporal period vary with E as E 0? Solution: In the small- x region where the particle is trapped the equation for conservation of energy may be approximated 1 2 m d 2 x dt 2 + U p 2 | x | (1- 1 4 | x | ) = E As in the text we define variables to represent the rescaled quantities which are to remain finite as the...
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