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Generalized Newtons Cradle American Journal of Physics 80, 1078 (2012); 83, 110 (2015) [editors pick]83, 110 (2015) editors pick Jungil Lee Korea University in collaboration with June-Haak Ee (Korea U.) KIAS Journal Club Meeting February 3, 2015

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Contents

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Introduction to 1D multiple collision These recurrence relations are analytically solvable. Jungil Lee 3 KIAS, 2015

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Theoretical backgrounds

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Conservation Laws 5 Elastic collision Jungil Lee KIAS, 2015

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Equivalence of the elastic collision and the rotational transformation 6 Jungil Lee KIAS, 2015

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Equivalence of elastic collision and rotational transformation 7 where,. Jungil Lee KIAS, 2015

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Equivalence of elastic collision and rotational transformation 8 Jungil Lee KIAS, 2015

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Collisions against a rigid wall 9 Jungil Lee KIAS, 2015

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10 Jungil Lee KIAS, 2015 Recurrence relation

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Collisions between two identical blocks 11 Jungil Lee KIAS, 2015

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Generalization from 2D to 3D 12 Jungil Lee KIAS, 2015

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Collisions between two identical blocks 13 Jungil Lee KIAS, 2015

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Analytic Solutions 1: Collisions against the wall

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Collisions against the wall 15 Jungil Lee KIAS, 2015

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Velocities of the block and the ball 16 Jungil Lee KIAS, 2015 where and

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Collision position 17 Jungil Lee KIAS, 2015 where and if

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Collision position By using a trigonometric identity 18 Jungil Lee KIAS, 2015

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Collision time 19 Jungil Lee KIAS, 2015

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The analytic solutions The closed-form solutions for the velocities, positions, and times of the ball and the block: 20 Jungil Lee KIAS, 2015

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Position as a function of time The position of the block: 21 Jungil Lee KIAS, 2015 The position of the ball:

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Continuum Limit 22 Jungil Lee KIAS, 2015 where

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Ee and Lee, Am. J. Phys. 80, 1078 (2012) The trajectory of the block. The black line represents the actual path of the block. The blue and red curves are the upper and lower bound of the paths, respectively. Dipole repulsion!! 23 Jungil Lee KIAS, 2015

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One-molecule piston system Analytic solutions for motion of one-molecule piston system Near the turning point, the ideal gas law holds: 26 Jungil Lee KIAS, 2015 Pressure on the piston Volume of the piston Temperature of gas molecule

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Analytic Solutions 2: Collisions of two identical blocks

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Collisions of two identical blocks 26 Jungil Lee KIAS, 2015

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Velocities of the participants 27 Jungil Lee KIAS, 2015 Participants beforeafterbeforeafter

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Total number of collisions 28 Jungil Lee KIAS, 2015

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Terminal velocities 29 Jungil Lee KIAS, 2015 Ee and Lee, Am. J. Phys. in press arXiv:1310.5200 [physics.class-ph]

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A Generalized Newtons Cradle 30 Jungil Lee KIAS, 2015

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Summary

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Thank you