Re: [math-fun] Verticals
"Adam P. Goucher" <apgoucher@gmx.com> wrote:
"Keith F. Lynch" <kfl@KeithLynch.net> wrote:
These ideas can of course be extended to lumpier worlds. Could a local vertical ever point directly away from the center?
Suppose the planet is a solid torus of uniform density. Then if you stand on the inner Equator, gravity will pull you directly away from the centre of the planet.
Right.
(If you insist that the planet is homotopy-equivalent to a closed ball, just include an infinitely thin membrane spanning the hole.)
My solution was a hollow sphere with a hole drilled in the north pole. A plumbbob at the inner south pole would point directly away from the center. I'm now exploring two related ideas, each of which I may turn into a new post: It's well known that the Moon's gravity is too lumpy for there to be any long-term stable orbits. Similarly around Mercury, where the first and only Mercury orbiter did in fact recently crash into the planet. Of course an orbit can be adjusted as necessary by expending rocket fuel, but you would soon run out. Would it be possible to maintain an orbit indefinitely by having a satellite consisting of two masses on a tether, using solar energy to reel in and out the tether to circularize the orbit? A flywheel may also be necessary. What about the opposite extreme, a perfectly fluid planet? What shape would such a planet have? If it wasn't rotating, it would obviously be a sphere. With rotation, would be it be an ellipsoid, or is that only a good first-order approximation where gravity greatly exceeds centrifugal force? Would the shape depend on the distribution of mass, e.g. would it be one shape if nearly all the mass was near the center, and a different shape if the planet was of uniform density?
Within a hollow spherical shell (uniform thickness and density), the gravitational forces cancel. The plumb Bob floats. On Sep 6, 2015 2:18 PM, "Keith F. Lynch" <kfl@keithlynch.net> wrote:
"Adam P. Goucher" <apgoucher@gmx.com> wrote:
"Keith F. Lynch" <kfl@KeithLynch.net> wrote:
These ideas can of course be extended to lumpier worlds. Could a local vertical ever point directly away from the center?
Suppose the planet is a solid torus of uniform density. Then if you stand on the inner Equator, gravity will pull you directly away from the centre of the planet.
Right.
(If you insist that the planet is homotopy-equivalent to a closed ball, just include an infinitely thin membrane spanning the hole.)
My solution was a hollow sphere with a hole drilled in the north pole. A plumbbob at the inner south pole would point directly away from the center.
I'm now exploring two related ideas, each of which I may turn into a new post:
It's well known that the Moon's gravity is too lumpy for there to be any long-term stable orbits. Similarly around Mercury, where the first and only Mercury orbiter did in fact recently crash into the planet. Of course an orbit can be adjusted as necessary by expending rocket fuel, but you would soon run out. Would it be possible to maintain an orbit indefinitely by having a satellite consisting of two masses on a tether, using solar energy to reel in and out the tether to circularize the orbit? A flywheel may also be necessary.
What about the opposite extreme, a perfectly fluid planet? What shape would such a planet have? If it wasn't rotating, it would obviously be a sphere. With rotation, would be it be an ellipsoid, or is that only a good first-order approximation where gravity greatly exceeds centrifugal force? Would the shape depend on the distribution of mass, e.g. would it be one shape if nearly all the mass was near the center, and a different shape if the planet was of uniform density?
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Bill: That's why Goucher drilled the hole in the pole. Never underestimate Goucher. --Bill On 2015-09-08 02:40, William R. Somsky wrote:
Within a hollow spherical shell (uniform thickness and density), the gravitational forces cancel. The plumb Bob floats. On Sep 6, 2015 2:18 PM, "Keith F. Lynch" <kfl@keithlynch.net> wrote:
"Adam P. Goucher" <apgoucher@gmx.com> wrote:
"Keith F. Lynch" <kfl@KeithLynch.net> wrote:
These ideas can of course be extended to lumpier worlds. Could a local vertical ever point directly away from the center?
Suppose the planet is a solid torus of uniform density. Then if you stand on the inner Equator, gravity will pull you directly away from the centre of the planet.
Right.
(If you insist that the planet is homotopy-equivalent to a closed ball, just include an infinitely thin membrane spanning the hole.)
My solution was a hollow sphere with a hole drilled in the north pole. A plumbbob at the inner south pole would point directly away from the center.
I'm now exploring two related ideas, each of which I may turn into a new post:
It's well known that the Moon's gravity is too lumpy for there to be any long-term stable orbits. Similarly around Mercury, where the first and only Mercury orbiter did in fact recently crash into the planet. Of course an orbit can be adjusted as necessary by expending rocket fuel, but you would soon run out. Would it be possible to maintain an orbit indefinitely by having a satellite consisting of two masses on a tether, using solar energy to reel in and out the tether to circularize the orbit? A flywheel may also be necessary.
What about the opposite extreme, a perfectly fluid planet? What shape would such a planet have? If it wasn't rotating, it would obviously be a sphere. With rotation, would be it be an ellipsoid, or is that only a good first-order approximation where gravity greatly exceeds centrifugal force? Would the shape depend on the distribution of mass, e.g. would it be one shape if nearly all the mass was near the center, and a different shape if the planet was of uniform density?
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participants (3)
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Keith F. Lynch -
rwg -
William R. Somsky