# hw4 - Physics 318 Problem Set 4 Due Wednesday 1 A particle...

This preview shows pages 1–2. Sign up to view the full content.

Physics 318: Problem Set 4 Due Wednesday, February 20, 2008 1. A particle is constrained to move without friction along the curve y = y ( x ) in two dimensions in a uniform gravitational field. It starts from rest at ( x 1 , y 1 ) and ends at ( x 2 , y 2 ), where y 1 = y ( x 1 ) and y 2 = y ( x 2 ). a. Show that the time taken to reach the second point is T [ y ] = integraldisplay x 2 x 1 dx radicalbig 1 + y ( x ) 2 radicalbig 2 g [ y 1 y ( x )] . b. Compute the choice of function y ( x ) which minimizes T [ y ] as follows. Use the fact that the integrand in T [ y ] does not depend on x to obtain a first integral of the Euler equation (ie to find a relation between y and y ). Integrate this differential equation using a substitution of the form y = k 2 sin 2 ( ϕ/ 2) h with suitable constants k and h . Show that the resulting curve is a cycloid [a curve given parametrically by x x 1 = a ( ϕ sin ϕ ) and y 1 y = a (1 cos ϕ ) for some constant a ] with a cusp at the point ( x 1 , y 1 ). 2. Generalize the derivation of the Euler equation for the function y ( x ) which minimizes the functional J [ y ] = integraltext x 2 x 1 dxF ( y, y , x

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern