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Unformatted text preview: Ph-507. Homework 2 (due: Friday, February 4).PROBLEM 2-1 (3 pts)Considetr a relativistic particle in an external potential,U=k|x|:L=−mc2r1−˙x2c2−k|x|Find the period of its oscillations as a function of amplitudex.PROBLEM 2-2 (2 pts)A "waterslide" has a shape given by the following 3D curve:x(z) =zcos (2πz/λ),y(z) =zsin (2πz/λ)Herethe positive direction ofzis taken to be downward, parameterλis unknown, and this spiral path makesNtotalturns (assumeNÀ1). Find the overall travel time down this slide, if the initial andfnal speeds are zero andv,respectively. Neglect dissipation.PROBLEM 2-3 (7 pts)A particle of massmis confned by potentialU(x). Find the period of its oscillations, as a function of the totalenergyE.(a)U(x) =Utan2(x/ξ);(b)U(x) =U£e−2x/ξ−2e−x/ξ¤;(c)U(x) =−Uh(x/ξ)2−1i2. Express the result in terms of elliptic integralK(z).PROBLEM 2-4 (3 pts)Due to the presence of ions, electrostatic interactions are normally "screened" in water. For example, electrostaticDue to the presence of ions, electrostatic interactions are normally "screened" in water....
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