Question

You can use the Euler-Lagrange equation to show how the shape of a suspended cable with no load is simply a catenary. However, if you suspend a much heavier load (i.e., a bridge) with the cable, the cable forms a parabolic shape. Does anyone know how to show this fact using the Euler-Lagrange equation?

Options:
A) The Euler-Lagrange equation describes the equilibrium shape of a cable under varying loads, demonstrating that under heavier loads, the equation transforms to a parabolic form.
B) Applying different boundary conditions in the Euler-Lagrange equation reveals that for heavier loads, the solution leads to a parabolic shape rather than a catenary.
C) Utilizing a modified potential energy function in the Euler-Lagrange equation accounts for the increased load, resulting in a parabolic cable shape instead of a catenary.
D) Altering the Lagrangian for a cable under varying loads in the Euler-Lagrange equation leads to a modified equation of motion, demonstrating the transition from a catenary to a parabolic shape under heavier loads.

Answer 1

The Euler-Lagrange equation can be altered to show the transition from a catenary shape to a parabolic shape under heavier loads, demonstrating the use of a modified equation of motion.

The Correct Option is; option D.

Step-by-step explanation:

The correct option to show how the shape of a suspended cable with no load is a catenary shape and transforms to a parabolic shape under a heavier load using the Euler-Lagrange equation is option D.

By altering the Lagrangian for a cable under varying loads in the Euler-Lagrange equation, we can obtain a modified equation of motion. This modified equation demonstrates the transition from a catenary shape to a parabolic shape under heavier loads.

This transformation occurs because the increased load on the cable changes the potential energy function, resulting in a parabolic cable shape instead of a catenary.