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SolnPS11

# SolnPS11 - PhysﬁOl/F 11/Pr0blem Set 11 Due Dec 5 2011...

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Unformatted text preview: PhysﬁOl/F 11/Pr0blem Set 11 Due Dec 5, 2011 -.:... 11.1H Normal modes A particle of mass in and charge q is connected to the origin by a Spring of constant 1110392. Crossed E and B fields are externally applied. E = E0XA and B = Bozn, Where EU and B0 are constants. 1. Write down the Lagrangian for the system assuming the motion is in the 2D X-y plane only. ' 2. Find a static equilibrium. What are the balancing forces for this? 3. Obtain the equations for small oscillations about this equilibrium. 4. Find the normal mode eigenfrequencies. Is the system stable? 5. What is the effect of E0 on your normal modes. 6. What are the approximate eigenfrequencies for B0 a» 0 and B0 —> on. Keep corrections up to the first non—vanishing order. Comment. 11.1G Rotating Frames Goldstein Ch4 Problem 4.24 11.11! Normal modes Three smaJi bells cf equal mass m and negligible radius a mauve ﬁithout ﬁ-iction inua. circuiar tunnel of radius R > c. There is no gravity. The belis are connected by springs, each of unstretcheci iength Length gar}? and spring constant 1:. At equilibrium the belts are in the positicms shown in the ﬁgure above. The tote] potential energy of the balls is v = in? ((9151 w M244 (d2 4 \$3)? '+ (<53 - M”): where d), is the angular displacement of bad} 2' from its equilibrium position. The halls are now perturbed from their equilibrium position . , (a) What are the eigenfrequeneiﬁ of osailletion and their degeneracint? {9 points] (Hint: The equation for eigenvalues, although cubic inprineiple, does not have the constant term. Thus the mats are easy to"ﬁnd) - - '- 83) Find and describe the eigenmodes corresponding to the frequencies you found' in (a)? ' [ 9 points} (:3) Note that the Lagrangian does nctdchzmge if the same angle is added to the three coordinates cm. This suggests that in this problem, in addition to energy, there is another conserved quantity. What is this constant ef the metien? ' [ 2 points} ...
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