4. Maximum Area You are given a string of fixed length l with one end fastened at the origin O, and you are to place the string in the ry plane with its other end on the x axis in such a way as to enclose the maximum area between the string and the x axis. Show that the required shape is a semicircle. Hint: The area enclosed is of course ydx, but show that you can rewrite this in the form fds, where s denotes the distance measured along the string from O, where f = yvI - y2, and y' denotes dy/ds. Since f does not involve the independent variable s explicitly, you can exploit the "second form" of Euler-Lagrange Equation.

4. Maximum Area You are given a string of fixed length l with one end fastened at the origin O, and you are to place the string in the ry plane with its other end on the x axis in such a way as to enclose the maximum area between the string and the x axis. Show that the required shape is a semicircle. Hint: The area enclosed is of course ydx, but show that you can rewrite this in the form fds, where s denotes the distance measured along the string from O, where f = yvI - y2, and y' denotes dy/ds. Since f does not involve the independent variable s explicitly, you can exploit the "second form" of Euler-Lagrange Equation.

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Question

4.
Maximum Area You are given a string of fixed length l with one end
fastened at the origin O, and you are to place the string in the ry plane with its other
end on the x axis in such a way as to enclose the maximum area between the string and
the x axis. Show that the required shape is a semicircle.
Hint: The area enclosed is of course ydx, but show that you can rewrite this in the
form fds, where s denotes the distance measured along the string from O, where
f = yvI - y2, and y' denotes dy/ds. Since f does not involve the independent variable
s explicitly, you can exploit the "second form" of Euler-Lagrange Equation.

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