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Consider this. I have an Earth-sized quantity of water that I throw into space. Naturally, it will assume the shape of a ball. Only if it is non-rotating and is not being influenced by a tidal field. So is there some sort of simple formula that ties these two properties together? Yes. It is called the equation of hydrostatic equlibrium. $$... 3 Far away from a black hole, spacetime is curved only a little bit, and many different things could curve it like that out there. It's like if you had a dollar in your pocket, and it's been there for a long time, and you can't remember if you got it from your boss or from your friend. But a dollar is a dollar. So you could have a massive star, or a black ... 11 Although we don't have a quantum theory of gravity, we think we have some reliable knowledge about the properties of black holes from general relativity. One thing we think we know is the so-called "No-hair conjecture", which says that black holes can be described by just three numbers: mass, charge, and angular momentum (i.e. how much they are spinning). ... 0 The singularity probably does not exist, as GR likely breaks down at those size / energy scales. When we have a full quantum description of gravity we may know what's really there. By the way, the part of the black hole we fully understand is actually the vacuum solution - the Schwarzschild metric - which includes the event horizon but not the source mass. ... 0 It's almost certainly incorrect that the center of a black hole is a singularity as this would be at odds with quantum mechanics. Just how exactly it looks like would be something to ask of a theory of quantum gravity! Regardless of being a singularity or not, the mass is determined by how much mass you stuff into your black hole. Hence black holes of ... 0 The density of black holes isn't infinite. Some black holes have the billionfold density of our sun (like the black holes in center of galaxies). There are big and small black holes. 0 Consider 2D (XY) case for simplicity, \vec x = x\vec e_x + y\vec e_y. \vec e_x is unit vector along x-axis. By definition, gradient is the following operator upon scalar: \nabla=(\frac{\partial}{\partial x}\vec{e_x} + \frac{\partial}{\partial y}\vec{e_y}) Apply this operator to \vec p\cdot\vec x: \nabla(\vec{p}\cdot \vec ... 0 It's not really related to your question but I've read that heavier white dwarfs are smaller than lighter white dwarfs and heavier neutron stars are thought to be smaller than lighter ones. When you get that much mass together, gravity tends to win. Even on the scale of the earth or Mercury, the planet's cores are crushed into greater density. I'm not ... 15 The reason is electron degeneracy pressure. The cores of giant planets are dense enough that the electrons in the gas occupy about h^3 of phase space each. The Pauli exclusion principle means that they cannot all occupy low energy/momentum states. This means that even at relatively cool temperatures the gas can still exert considerable pressure due to the ... 1 The relationship between the density \rho of some quantity Q and its flux \mathbf j is always in the form of a continuity equation,$$ \frac{\partial \rho}{\partial t}+\nabla\cdot\mathbf j=S,  where $S$ is a source term for $Q$, equal to the amount of $Q$ that appears per unit volume per unit time at each position. If $Q$ is conserved and you've ...

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Two factors: Overall density. A boat isn't only metal. It contains copious amounts of plastics and wood, which have lower densities than water. You have to calculate the overall density of the object, not just the metal. Shape: If boats were made in a sink-friendly shape, they would certainly sink (regardless of wood and all). Engineers design boats to be ...

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This is because the whole boat, along with the air in the boat, is lighter than the water it displaces. For example, if a small boat will take up 1 cubic meter of water, then it has to be heavier than the weight of 1 cubic meter of water. This is explained in this post by What If here. For the same reason that bowling balls float (because salt water the ...

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It seems like they ought to sink because we're used to seeing things fall. But for the ship to sink it has to push aside some water, which has nowhere to go but up. So it's a question: does the ship 'want' to sink more than the water 'wants' not to rise? It turns out that just depends on whether the ship weighs more or less than the amount of water that ...

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