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a11f02

# a11f02 - l 1 = l 2 = l φ 1 = φ and φ 2 = ψ 1 Deﬁne...

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INTERMEDIATE MECHANICS II — PHY 4936 HOME AND CLASS WORK – SET 2 (September 9, 2011) Assignments 6 and 7 are motivated by Feynman’s text: (6) Show that “the mean square of something that deviates around an average ... is always greater that the square of the mean” (4 points). Due in class (4 points). (7) What is a 3D conservative force? Write down three definitions (3 points). Due in class. – Read Landau-Lifshitz p.11 and p.12. Due September 14 before class. – Read Landau-Lifshitz § 6, § 7, § 8. Due September 16 before class. (8) A double pendulum consists of two simple pendula, with one pendulum sus- pended from the bob of the other. Assume that the two pendula have equal lengths, have bobs of equal mass and are confined to move in the same plane. (Compare problem 1 of Landau-Lifshitz with
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Unformatted text preview: l 1 = l 2 = l , φ 1 = φ and φ 2 = ψ .) 1. Deﬁne angles φ and ψ for the pendula with respect to the gravity direction and write down the Lagrange function. 2. Derive the equations of motion for small oscillations around the rest posi-tion φ = ψ = 0. (Small oscillations neglect all terms in the Taylor expan-sion of the Lagrangian, which are higher than quadratic in combinations of φ, ˙ φ, ψ, ˙ ψ .) Due Friday, September 16 before class (10 points). (9a) Assume a bilinear kinetic Energy T = X j,k a jk ˙ q j ˙ q k and prove X i ˙ q i ∂T ∂ ˙ q i = 2 T . Due in class (2 points). (9b) Generalized Momentum: Calculate ∂L ∂ ˙ x i , i = 1 , 2 , 3 , for L = 1 2 m~v 2-V ( ~x ) . Due in class (2 points)....
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