Can a catenary be a parabola? In one of my text books (Thomas' Calculus, I think), one of the exercises asked us to find the arc length of a hanging wire that was weighted such that it traced a parabola, or something to that effect. Point is, it wanted us to find the arc length of a parabola, and tried to justify it with a real world example.
Now, I know that, in general, a hanging wire forms a catenary, not a parabola, so my question is: is it possible for a catenary to be a parabola, at least within a given finite width?

No, it is not possible. For instance, take a point $P$ of a parabola. Consider the straight line $l_P$ passing through $P$ parallel to the axis of symmetry of the parabola. Let $l_P'$ be the reflection of $l$ on the normal to the parabola at $P$. Then all these straight lines $l_P'$ have a point in common (which is the focus of the parabola). The catenary doesn't have this property.
