A point mass m slides without friction from O = (0,0) to P = (a, b) on a curve C under the action of constant gravity (see Figure) with vanishing initial velocity. The time it takes for m to reach P is given by P 1 -ds, Jo T = where ds = V(dr)² + (dy)² and v is the speed. The goal is P=(a,b) y to find the curve C that minimises T. (a) Write T as T = Sº F[y(x), y'(x)]dx and determine the function F. (b) Making use of F – y = const (see Problem 1), derive the relation y' y ƏF dy' V# - 1, where d is a constant. (c) Use the parametric representation y(@) = d sin² = $(1 – cos ø) and determine r(6). %3D The extremal curve is a cycloid.

A point mass m slides without friction from O = (0,0) to P = (a, b) on a curve C under the action of constant gravity (see Figure) with vanishing initial velocity. The time it takes for m to reach P is given by P 1 -ds, Jo T = where ds = V(dr)² + (dy)² and v is the speed. The goal is P=(a,b) y to find the curve C that minimises T. (a) Write T as T = Sº F[y(x), y'(x)]dx and determine the function F. (b) Making use of F – y = const (see Problem 1), derive the relation y' y ƏF dy' V# - 1, where d is a constant. (c) Use the parametric representation y(@) = d sin² = $(1 – cos ø) and determine r(6). %3D The extremal curve is a cycloid.

University Physics Volume 1

18th Edition

ISBN:9781938168277

Author:William Moebs, Samuel J. Ling, Jeff Sanny

Publisher:William Moebs, Samuel J. Ling, Jeff Sanny

Chapter2: Vectors

Section: Chapter Questions

Problem 2.10CYU: Check Your Understanding Verify that vector v V obtained in Example 2.14 is indeed a unit vector by...

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