prelim1999

prelim1999 - Table of Contents Complete Partially Complete...

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Unformatted text preview: Table of Contents Complete Partially Complete Not Complete Preliminary Examination 1999 2 Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Problem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Problem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Problem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Problem 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Problem 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Problem 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Problem 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Problem 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Problem 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Problem 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Problem 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Problem 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Problem 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1 Preliminary Examination September , 1999 1. Consider a yo-yo climbing its string. Assume that the string is vertical. Notation: r = radius of yo-yo axle m = mass of yo-yo I = moment of inertia of yo-yo g = acceleration of gravity = angular velocity of yo-yo about its axis Show that the equations of motion for the yo-yo reduce to an equation for in terms of r , m , I , and g (and nothing else). Show that your equation implies that the total energy of the yo-yo is conserved as it climbs. Why must that be true? Solution (a) Incomplete: Need to answer energy question.] y th r PSfrag replacements y r The kinetic energy of our yo-yo is given by T = 1 2 m y 2 + 1 2 I 2 Because positive y is in the downward direction, the potential energy is V = mgy Thus, the Lagrangian is L = 1 2 m y 2 + 1 2 I 2 + mgy Also, we note that our system is under the following constraint: f ( y, ) = y r = 0 y = r y = r Thus, the Euler-Lagrange equations are L y d d t L y + f y = 0 mg m y + = 0 = m ( g y ) L d d t L + f = 0 I + ( r ) = 0 I + mr ( g y...
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prelim1999 - Table of Contents Complete Partially Complete...

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