Jerry Avins wrote: (snip of orbits inside a tunnel)> The heavy core distorts the nice theoretical linear decline. In fact, > gravity will continue to increase with depth until the core is nearly > reached if the core is dense enough and the rest is fluffy enough.> The period is 2*pi*sqrt(r/g) When g = kr (k constant) the period is > independent of r. Like turtles, it's 88 minutes all the way down.Yes. I ended up with an equation with an R**3 in it, that is, the planet radius, the right equation, and concluded that it was just like Kepler, not noticing R vs. r. (Well, it was late at night. The fun of take home closed book quizzes.) -- glen
Rate Gyros and Accelerometers
Started by ●April 22, 2007
Reply by ●April 26, 20072007-04-26
Reply by ●April 27, 20072007-04-27
glen herrmannsfeldt wrote:> Jerry Avins wrote: > (snip of orbits inside a tunnel) > >> The heavy core distorts the nice theoretical linear decline. In fact, >> gravity will continue to increase with depth until the core is nearly >> reached if the core is dense enough and the rest is fluffy enough. > >> The period is 2*pi*sqrt(r/g) When g = kr (k constant) the period is >> independent of r. Like turtles, it's 88 minutes all the way down. > > Yes. I ended up with an equation with an R**3 in it, > that is, the planet radius, the right equation, and concluded > that it was just like Kepler, not noticing R vs. r. > (Well, it was late at night. The fun of take home closed > book quizzes.)I shouldn't have had to think about your problem, but I did. In the 50s, there was talk of long-distance transportation tunnels that would be bored on a straight path. A tunnel from New York to San Francisco, for example, would be far below the surface at its midpoint. Despite the tunnel's being straight, it would seem to be slanted downward at each end. If evacuated to eliminate air resistance, the gravity-driven end-to-end time would be 88 minutes regardless of the separation of the destinations. "I shoulda known!" Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Reply by ●April 27, 20072007-04-27
Jerry Avins <jya@ieee.org> writes:> Heinrich Wolf wrote: >> Jerry Avins <jya@ieee.org> writes: >>... >>>... > >> But the reason why I reply is a related problem which you might be >> interested in: provided the earth were a homogenous sphere, what would^^^^^^^^^^^^^^^^^>> be the pressure in the center of the earth? Many books on geology >> seem to get this wrong: they don't consider the reduction of gravity >> when moving inwards. I did the (rather simple) calculation and got an >> even greater result. Though baffled at the first moment, I saw why: >> when doing it right, the load to a surface element at the center of >> the earth isn't a prisme, but a cone. > > did your calculation account for the likelihood that the core is much > denser than the mantle?See the carrets above. Please correct me, if this doesn't answer your question. (No native english speaker.)> >> BTW Jerry: Ever heared about M.Schuler? This guy might have proposed >> something similar to Avin's stable platform already before 1940. > > I don't know of Schuler's work. Avins's stable platform is a parody of > Draper's dreamed up to explain on physical grounds what he worked out > with equations. I think Draper actually wrote a paper called "The > Eighty-eight Minute Pendulum". The parody didn't require deep thought.Parody--- I understood it this way. Schuler's work seems to be related to the gyro-compass; have no good refernce yet, but if I find more I will ping you in this NG. -- hw
Reply by ●April 27, 20072007-04-27
Jerry Avins <jya@ieee.org> writes:> glen herrmannsfeldt wrote: >> Jerry Avins wrote: > ... > I shouldn't have had to think about your problem, but I did. In the > 50s, there was talk of long-distance transportation tunnels that would > be bored on a straight path. A tunnel from New York to San Francisco, > for example, would be far below the surface at its midpoint. Despite > the tunnel's being straight, it would seem to be slanted downward at > each end. If evacuated to eliminate air resistance, the gravity-driven > end-to-end time would be 88 minutes regardless of the separation of > the destinations. "I shoulda known!"I was not aware of this wonderful property of rotational symmetric harmonic oszillators up to this moment. It's so easy to see: a) For any harmonic oszilator period is independent of amplitude. b) Given an n-dimensional harmonic oszillator, then you can define a function of a single variable, by giving (n-1) variables a fixed value. Now the functions defined this way for various choices of the fixed coordinates obviously differ only by a constant. Geometrically: if you have a rotational parabola, then intersections with a plane parallel to the axis of rotation will show _congruent_ parabolas. That's it. -- hw
Reply by ●April 27, 20072007-04-27
Heinrich Wolf wrote:> Jerry Avins <jya@ieee.org> writes: > >> Heinrich Wolf wrote: >>> Jerry Avins <jya@ieee.org> writes: >>> ... >>>> ... >>> But the reason why I reply is a related problem which you might be >>> interested in: provided the earth were a homogenous sphere, what would > ^^^^^^^^^^^^^^^^^ > >>> be the pressure in the center of the earth? Many books on geology >>> seem to get this wrong: they don't consider the reduction of gravity >>> when moving inwards. I did the (rather simple) calculation and got an >>> even greater result. Though baffled at the first moment, I saw why: >>> when doing it right, the load to a surface element at the center of >>> the earth isn't a prisme, but a cone. >> did your calculation account for the likelihood that the core is much >> denser than the mantle? > > See the carrets above. Please correct me, if this doesn't answer > your question. (No native english speaker.)Careless reading. Sorry!>>> BTW Jerry: Ever heared about M.Schuler? This guy might have proposed >>> something similar to Avin's stable platform already before 1940. >> I don't know of Schuler's work. Avins's stable platform is a parody of >> Draper's dreamed up to explain on physical grounds what he worked out >> with equations. I think Draper actually wrote a paper called "The >> Eighty-eight Minute Pendulum". The parody didn't require deep thought. > > Parody--- I understood it this way. Schuler's work seems to be > related to the gyro-compass; have no good refernce yet, but if I find > more I will ping you in this NG.Thank you. Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Reply by ●April 27, 20072007-04-27
Heinrich Wolf wrote:> Jerry Avins <jya@ieee.org> writes: > >> glen herrmannsfeldt wrote: >>> Jerry Avins wrote: >> ... >> I shouldn't have had to think about your problem, but I did. In the >> 50s, there was talk of long-distance transportation tunnels that would >> be bored on a straight path. A tunnel from New York to San Francisco, >> for example, would be far below the surface at its midpoint. Despite >> the tunnel's being straight, it would seem to be slanted downward at >> each end. If evacuated to eliminate air resistance, the gravity-driven >> end-to-end time would be 88 minutes regardless of the separation of >> the destinations. "I shoulda known!" > > I was not aware of this wonderful property of rotational symmetric > harmonic oszillators up to this moment. It's so easy to see: > > a) For any harmonic oszilator period is independent of amplitude. > > b) Given an n-dimensional harmonic oszillator, then you can define a > function of a single variable, by giving (n-1) variables a fixed > value. Now the functions defined this way for various choices of the > fixed coordinates obviously differ only by a constant. Geometrically: > if you have a rotational parabola, then intersections with a plane > parallel to the axis of rotation will show _congruent_ parabolas. > That's it.But the straight tunnel is neither the fastest nor the slowest gravity-powered path between two cities. Steeper initial and final grades can yield shorter times. The curve degenerates to a cycloidal path in a uniform gravitational field. Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
