Assignment 7 Solved - Mechanics | PHYSICS 311, Assignments of Mechanics

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6-1. If we use the varied function y(a,x) = x +a sin m(1-x) (1) Then, dy = 1 - an cos (1-x) (2) Thus, the total length of the path is L s=f /1+ GQ) ax 0 1 = f [2 - 2am cos w(1 - x)satn® cos? m(1 - x)]?/? dx (3) 0 Setting m(1-x) = u, the expression for S becomes T sel J v2 [1 - an cos u + 5 a?n? cos? uj?/? gy (4) 7 0 The integral cannot be performed directly since it is, in fact, an elliptic integral. Because a is a small quantity, we can expand the integrand and obtain 1 S= u f fl - 3 (am cos u - 3 a?m? cos? u) - 3 (am cos u - 3 a?n? cos* u)® + eee] du (5) If we keep the terms up to cos? u and perform the integration, we find S=yv2+ ui wa? (6) which gives 8S. V2 ata (7) Therefore, as ’a a=0 7° ® and S is a minimum when a = 0.