show/hide this revision's text 3 same idea, simplified

EDIT: More briefly than before . Here is a "physics" "proof": Far naive physical argument which might meet the request of a point of view from any gravitational forces, which the napkin ring sits captured result would immediately appear to be just what one would expect before going through the argument.

  • A blob of (incompressible) fluid volume V will form a spherical ball of radius (what it needs to be) if uncontrained
  • A blob of fluid volume V constrained between two parallel rigid glass plates at z=r and z=-r . The will form the height 2r cylindrical center is also made central slice of rigid glass. However the curved outer surface is a flexible membrane and the interior sphere of the napkin ring radius R where R is filled with just right so that the appropriate slice has volume of incompressible water. Even though the membrane V (provided r is free not too large relative to move (other than being fixed at the plates) the water will hold the napkin ring shape (the portion of V in which case we get a sphere)
  • A blob of fluid of volume V constrained between the plates). Why? Imagine extending the membrane to be two parallel plates at z=r and z=-r and with a full sphere cylinder of height 2r and radius R, filling q imposed in the previously empty cylinder middle will form (and caps) along with water and then having the glass disappear. The fluid will hold its spherical shape due to fluid pressure. The portion of water which was in cylinder) the napkin ring won't know height 2r central slice of a sphere of radius R where R is just right so that anything changed, the glass exerted the same forces as the new water does along the same contourslice has volume $V+\pi q^2 h$. This shows that the shape of might not have the napkin ring is completely determined by curved boundry reach the volume of water it encloses and not by R (so long as R>r.). Since this cylinder
  • Imagine that the volume is independent of R, we might as well use R=rjust right to get that napkin ring. QED?

    Variation: Start with a volume of $4/3 \pi r^3$ of fluid in a membraneNow start shrinking q. It We will naturally take still have a spherical shapenapkin ring. Imagine a long cylinder of glass (inner radius 0) passing along the axis Keep going until q=0 and glass plates tangent at the poles. Now let the radius of we see that the centeral cylinder grow. The water always thinks it is part of some bigger volume was that of water and keeps a spherical shape sphere of radius r.

    • (since the glass presses back on the water with the force that read the water presses on it).previous version if you wish, it might not be worth it)
    show/hide this revision's text 2 spelling

    Here is a "physics" "proof": Far from any gravitational forces, the napkin ring sits captured between two parallel rigid glass plates at z=r and z=-r. The height 2r cylindrical center is also made of rigid glass. However the curved outer surface is a flexible membrane and the interior of the napkin ring is filled with the appropriate volume of incompressible water. Even though the membrane is free to move (other than being fixed at the plates) the water will hold the napkin ring shape (the portion of a sphere between the plates). Why? Imagine extending the membrane to be a full sphere of radius R, filling the previously empty cylinder (and caps) with water and then having the glass disappear. The fluid will hold its spherical shape due to fluid pressure. The portion of water which was in the napkin ring won't know that anything changed, the glass exerted the same forces as the new water does along the same contour. This shows that the shape of the napkin ring is completely determined by the volume of water it encloses and not by R (so long as R>r.). Since this volume is independent of R, we might as well use R=r. QED?

    Variation: Start with a volume of $4/3 \pi r^3$ of fluid in a membrane. It will naturally take a spherical shape. Imagine a long cylinder of glass (inner radius 0) passing along the axis and glagg glass plates tangent at the poles. Now let the radius of the centeral cylinder grow. The water always thinks it is part of some bigger volume of water and keeps a spherical shape (since the glass presses back on the water with the force that the water presses on it).
    show/hide this revision's text 1

    Here is a "physics" "proof": Far from any gravitational forces, the napkin ring sits captured between two parallel rigid glass plates at z=r and z=-r. The height 2r cylindrical center is also made of rigid glass. However the curved outer surface is a flexible membrane and the interior of the napkin ring is filled with the appropriate volume of incompressible water. Even though the membrane is free to move (other than being fixed at the plates) the water will hold the napkin ring shape (the portion of a sphere between the plates). Why? Imagine extending the membrane to be a full sphere of radius R, filling the previously empty cylinder (and caps) with water and then having the glass disappear. The fluid will hold its spherical shape due to fluid pressure. The portion of water which was in the napkin ring won't know that anything changed, the glass exerted the same forces as the new water does along the same contour. This shows that the shape of the napkin ring is completely determined by the volume of water it encloses and not by R (so long as R>r.). Since this volume is independent of R, we might as well use R=r. QED?

    Variation: Start with a volume of $4/3 \pi r^3$ of fluid in a membrane. It will naturally take a spherical shape. Imagine a long cylinder of glass (inner radius 0) passing along the axis and glagg plates tangent at the poles. Now let the radius of the centeral cylinder grow. The water always thinks it is part of some bigger volume of water and keeps a spherical shape (since the glass presses back on the water with the force that the water presses on it).