Question for the Physics-Minded

[beamjockey]

Recently I bought a new can of coffee.

I removed its plastic lid, and admired the shiny foil seal. On one side, ordinary atmospheric pressure. On the other, "vacuum." Its rim is attached to the circular rim of the can. The forces on it balance into a smooth convex form.


I began to wonder:

What shape is this?

Paraboloid? Section of a sphere?

I'm thinking it's a catenary of rotation. Am I right?

Comments

[kevinnickerson.livejournal.com]

It has to be a rotated catenary, but isn't that the same as a parabola?

(goes off to check the always there wikipedia)
Hm, that says it's not a parabola, but a hyperbolic cosine.

Ah well, not my area of expertise.

[del-c.livejournal.com]

Sufficiently shallow catenaries, like suspension bridges, look a lot like parabolas, but then they also look a lot like an arc of a circle. Catenaries are always confined within an arc and a parabola, and the three shapes converge on each other as they get shallower.

The converging on an arc makes sense, in terms of this discussion of pressure acting normal to the surface, while gravity acts down. As the arc gets shallower, the distinction between "down" and "normal to the curve" gets less important.

[mmcirvin.livejournal.com]

The difference is most visible in the tails. A parabola is a quadratic curve, with a constant second derivative, while a catenary's outer reaches are exponential, with all derivatives basically proportional to its own height.

So for a hanging cable or chain, you get a parabola in the case where the force is the same everywhere (as in an ideal suspension bridge, where the load is evenly distributed and the cable's weight and stiffness are negligible by comparison), and you get a catenary in the case where the cable is supporting its own weight, which increases with the supported length of cable.

But in both cases, we're talking about weight, not gas pressure.