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

[mmcirvin.livejournal.com]

If you model the foil as a linearly elastic membrane of negligible weight, I'd guess it should be a spherical cap. The curvature at any point should be proportional to the pressure difference.

[brooksmoses]

Yup, exactly. The critical distinction between this and a catenary is that in a catenary the force is always in the same direction, whereas with pressure differences as here the force is always normal to the surface.

[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.

[wcg.livejournal.com]

Yes, you're right.

[von-krag.livejournal.com]

OK we got the math out of the way, just how is the coffee "new and improved"?

[beamjockey.livejournal.com]

My guess? Beans that have never been picked before, improved by grinding.

I don't think the coffee now has added stannous fluoride, or anything.

[neowolf2.livejournal.com]

Civit-processed beans.

[eub.livejournal.com]

If the foil were soap-bubble, it would be a sphere cap... how does the foil behave? From the picture it looks like it may stretch by deforming pleats of some kind? (Inelastic deformation?)

Thought experiment to me suggests not a catenary: make the hole smaller and increase the pressure differential; does the tangent at the hole lip eventually go past vertical? I'd expect yes, which a catenary can't satisfy.

[mmcirvin.livejournal.com]

I don't get why it's a rotated catenary.

Suppose the pressure were greater on the inside, and suppose the membrane were a thin sheet of rubber. If you really jacked up the interior pressure, wouldn't it inflate into a spherical balloon? Or is this not the right model to use?

[mmcirvin.livejournal.com]

Hmm. I guess not a sphere, because this isn't a soap bubble; the material can't redistribute itself and will be stretched out more in the middle than on the edges. But definitely not a catenary.

[erik]

I agree with those saying it's got to be a section of a sphere unless the membrane is unevenly elastic. The deformation is caused by pressure, and the pressure is going to be uniform, so the deformation has to be uniform unless the elasticity is not.

[harvey-rrit.livejournal.com]

I've read all these comments, and I'm certain you must be right. If it were a paraboloidal section, extending it far enough would make it come to an ogive point in the middle.

(I had to visualize somebody on Mercury blowing lead bubbles to figure this out.)

[del-c.livejournal.com]

Mostly spherical, except at the wrinkly edge (not applicable in your can. A catenary (or whatever you call its domed equivalent) happens when the forces acting on the membrane have to pass along the membrane, but the external forces on each point of the membrane are all parallel (gravity). This isn't the case with pressure, where the forces acting on each point in the membrane are all normal to the membrane (air pressure).

So the dome of the Pantheon in Rome (if made of equal masses everywhere and built the right way) would be whatever you call a catenoid dome upside down, dominated by its own weight, but the classic science fiction domes on the Moon and Mars are rightly spherical, dominated by internal pressure.

[major-clanger.livejournal.com]

Apparently this is what's termed the Hencky-Campbell problem, and has been thought about for nearly a century. This paper from the USAF Research Laboratory at Kirtland AFB goes over the mathematics in eye-watering detail. The problem was also considered by NASA at around the same time. The answer seems to be that the curve is not a simple function but rather a power series that depends on factors including the elasticity of the membrane.

The practical application is using Mylar or similar reflective foil to make a cheap near-parabolic mirror.

UK amateur astronomer Maurice Gavin - who I knew back in the early 1980s as founder of an APA for swapping home computer programs for astronomy - did some experiments in this line. The results were optically awful but adequate for simple photometry.

[neowolf2.livejournal.com]

I think approaches like this have been used to make mirrors for concentrating solar power. Optical perfection is not required for that.

[stickmaker.livejournal.com]

This made me think of the corrector plate for a Schmidt-Cassegrain telescope. The first one of those was made by sealing a sheet of optical glass across the mouth of an old kettle, pulling a vacuum inside, polishing the glass flat, and releasing the vacuum.