Variational Calculus - variable density line

[Orion_PKFD]

Consider a line of length $L=\frac{\pi}{2}a$. We want to put small particles of lead (total mass of all particles M) in order that the line is hang in a circular arc. Both ends are at the same height. Show that the mass distribution needs to be

$\rho(y)=\frac{M}{2}\frac{a}{y^2}$

This exercise if different of the "usual" from textbooks because here we know the curve, but not the density. Anyone has an ideia in order to solve this?

Best regards!

 

[hunt_mat]

What is "y" and what do you think that you should minimise? We will help you but we won't solve the problem for you.

 

[Orion_PKFD]

"Oy" is the vertical axis. But it is reasonable that you can put the origin wherever you want. I would say that it should be helpful to place the origin in one of the endpoints.

We want to minimize the potential energy, U, but we know the shape of the curve. We need to find $\rho(y)$ in order that it keeps the shape (minimum U)