Liquid Planet

Classical Mechanics Level 1

A large spherical planet is made entirely of an incompressible liquid. What is the pressure at the center of this planet?

It is minimum at the center as the force from every direction would cancel out It is the same as everywhere else because density is uniform throughout the planet It is maximum at the center because liquid is being pulled towards the center due to gravity

Pranshu Gaba by Pranshu Gaba , 22, India

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9 solutions

Zain Majumder
Sep 9, 2018

Instead of considering a planet of liquid, consider a body of water. Pressure is higher at the bottom of the ocean than the surface of the ocean due to the water above applying pressure to the lower layers. The same logic applies to the planet of liquid. The force of gravity increases the pressure at the center of the planet.

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The planet has water both above and below the centre. Moreover, the point where the pressure is measured is the centre of gravity. Thus, the situation of a water body and that of this question are entirely different. How then can you draw this conclusion?

Jill Jani - 2 years, 9 months ago

Doesn't the fact that the liquid is incompressible mean that the pressure is the same everywhere?

José Salgado - 2 years, 9 months ago

What is the point of saying the liquid is incompressible? If the liquid is incompressible, does that mean that its density does not change under pressure?

Kermit Rose - 2 years, 9 months ago

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That's my question too. I hope they answer.

Jenny Stephens - 2 years, 9 months ago

I think saying the liquid was incompressible was misleading because the force of gravity would cause the liquid at the center of the planet to increase in density i.e. compress. If the premise that the liquid was incompressible was true then this would not be the case.

Brooks King - 2 years, 9 months ago

Your answer is correct if you assume gravitational force (or more precisely acceleration due to gravity, "g") to be constant even as you move to the center of the planet. However as per Newton's Law of gravitation, "g" varies proportionally to mass of the object and inversely to the square of the distance ("r") between the point under consideration and the object's center (here the planet's center). Here as the given point is inside the planet, the mass of the planet that exerts gravitational force is only going to be the portion which is present in the sphere of radius ("r") equal to the distance between the given point and the planet's center. The mass ("m", exerting the gravitational force) will thus be density of liquid times the volume of the sphere (4πr³/3). In other words, this mass "m" will vary as the cube of "r", and since "g" already varies directly with "m" and inversely with square of "r", therefore, in this case, "g" will be directly proportional to "r". So acceleration due to gravity "g" should be 0 at r=0, that at the center of the planet g=0. Since pressure P= (density of liquid)×g×(depth "h" of the given point below liquid surface), shouldn't pressure P=0 at the center as g=0 at the center even though depth "h" will be maximum?

Darshan Vyas - 2 years, 8 months ago
Michael Mendrin
Sep 9, 2018

Let's imagine that there is a plane gap that divides the liquid planet in half. While it's true that one at the center of the planet will experience a net gravitational pull of 0 0 because the contributions of each half cancels out, nevertheless there is enormous gravitational attraction between the two halves, and you'd be in the middle of it if you were at the center.

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How does this explain that pressure is greatest at the centre. Also, there is still no force at the centre. Shouldn't the pressure be zero?

Jill Jani - 2 years, 9 months ago

No...it won't be 0 at center. Consider a magnet. An another magnet is attracted to it,and gets attached. So,there wont be 0 force applied on it.

Rigved Naukarkar - 2 years, 9 months ago

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But both the magnets attract each other and not just one attracts the other. Moreover, do we conclude that there is a non zero force in the middle because they move towards each other

Jill Jani - 2 years, 9 months ago

yes, you're right. the pressure should be maximum at center. i thought later and surprized why i made such big mistake.

Brian Wright - 2 years, 9 months ago

Gravity was not mentioned or that it was assumed.

Jim Born - 2 years, 9 months ago

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water exerts gravitational pull

Michael Mendrin - 2 years, 9 months ago
Pietro Galvagni
Sep 10, 2018

Someone tell me that the center of the Earth is actually melted because of the enormous pressure that keeps the rocks at a liquid state

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The YouTube channel Veritassium talked about it on their episode "Is glass liquid". Even though the iron core of the earth is extremely hot, due it's enormous pressure it remains solid. It also explain the reason behind even though the mantle is solid, the huge pressure makes it behave like a liquid. (My English is still a work in progress, sorry about any mistake.)

Matheus Torres - 2 years, 9 months ago

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I believe it is hot at the centre of the Earth, not due to the enormous pressure but due to radioactive decay of radioactive elements. The heat generated has difficulty escaping all the way to the surface. Hence the high temperature.

Desmond Campbell - 2 years, 7 months ago

Where is the downvote button? Lol! This is a solution? With upvotes!

Chris Arsenault - 2 years, 8 months ago
Leon Green
Sep 9, 2018

Everything that has matter has a gravitational pull even liquid, it is just not as strong.

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Not as strong as what? The strength of gravity is relative to mass, not composition. This is why the sun, made up virtually entirely of gas, exerts such huge gravitational pull that holds the entire solar system in its orbit.

John Spink - 2 years, 9 months ago
Andrey Zharkikh
Sep 16, 2018

I guessed it right, but! If I try to describe it mathematically, pressure is the cumulative weight (taking into account decreasing gravity for the inner layers) per area of the surface surrounding the center, at some distance from it. P = F / A. Reducing the distance between the center and the surface slightly increases F, but relatively greatly decreases A, approaching zero, and shooting P up to infinity. What's wrong with this logic? Why do they say it's only about 365 GPa?

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Sudha D
Sep 15, 2018

Assuming there are no external forces but only internal force, the shape of round sphere is justified. My Physics is poor, so here's my open question to all: Lets consider earth, if gravitational force is so prominent, why aren't we all sucked into the center of earth crust. Or how things dont collapse make deep holes? And if gravity is the reason earth is round( slightly round), why is moon round? Why other plants are kind of round?

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Considering this planet is made up of incompressible liquid, it can be considered in the same way you would consider a whole solid ball. Thus, the centre of the ball would also be the centre of gravity and have the greatest pull.

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Inksa Inkeroinen
Sep 10, 2018

Hydrostatic pressure depends on depth of the liquid. The greatest depth is in the center of the planet, so it maximizes the pressure there.

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Planet is made entirely of incompressible liquid, the factor imply that liquid avoiding spread out applying by the force,gravity etc. The shape tend to be sphere instead of cube or other else., only sphere satisfied with condition that averagely strong of the gravity as same as anywhere on the surface of planet, because the force depend on surface of planet distance to the center of this planet. The liquid is incompressible that all liquid above the centre of planet pressure, so the deeper liquid,the more liquid ,the biggger pressure

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Correct me If I'm wrong.

It took me a while to realize why I made a mistake (I thought that there is minimal pressure due to cancelled forces) and why I think all the answers so far fails to answer properly why it is not 0 (near zero). I think what is missing here (to complete the explanation) is, that the force in pressure (pressure = force applied on a surface) DOES NOT act on TWO SIDES. That surface in the formula/definition has one side only. So for example If I put two equal magnets on a ferromagnetic coin - both magnets are attracted to the coin with equal force with opposite direction but the pressure is not 0, because we either (by definition) talk about one side or the other respectively. That's the reason why the pressure in the center of the planet is (by definition) biggest, because we always talk about a single sided surface, so there is no force cancellation.

Vladimir Fekete - 2 years, 9 months ago

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inspire by the idea that magnets on a ferromagnetic coin, very cool example , I am appreciate it. Refer to sphere,pressure applied on the surface through the center extension line. The remaining problems are not clear,correct me If I'm wrong： If conside Einstein's theory of time and space,what does the pressure or force apply to centre of planet come from?
if anything inspire you or me which come from our corresponse, it is a amazing.

丁大喵 by丁丁猫 - 2 years, 9 months ago

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I'm not sure what Einsteins' special theory of relativity has to do with this problem, I guess you meant Einsteins' general theory of relativity. Even if so, the GTR is unnecessary complication imho.

All we need is Newtons' laws of gravity (any Einsteins' laws are just correction, but it does not mean, that Newtons' laws are incorrect. They are just correct on smaller scale, or smaller order, if you wish).

We are talking about the pressure. Pressure is defined as: "Pressure is the amount of force applied perpendicular to the surface of an object per unit area." (wikipedia). So if you ask what is the source of the force, then it is mass of the soil directly above measured area. We expect the measuring device to be a very small outer surface of a sphere located in the center of the planet. Hence the source of the force is mass of the whole planet and that's the reason why p = F / S is so huge, because F comes from the huge mass of the planet and S is infinitesimally small.

Vladimir Fekete - 2 years, 9 months ago

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@Vladimir Fekete – "Hence the source of the force is mass of the whole planet and that's the reason why p = F / S is so huge, because F comes from the huge mass of the planet and S is infinitesimally small." The planet is so huge that center of mass pressure increase rapidly , it is easy to lead s->0,F->∞ black-hole consequence? I have no idea of it especially the physics problem like this, jump into my mind.

丁大喵 by丁丁猫 - 2 years, 9 months ago

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@丁大喵 By丁丁猫 – well there are (as always) two things: theory and real life. Yes, if you go to 0 with surface area using given formula you must end up with infinite pressure. But this is consequence of mathematics not physics. If we want to talk about physics, you need to also specify the range where laws you use operates/works/were deduced for.

In the world of physics a hypothesis has no meaning without experimental proof. So while you can theoretically calculate a pressure to be infinite with surface area going to zero, it does not mean that an experimental device attached to a given very small surface would measure a rectangular hyperbola with decreasing surface area (~ 1 / Area). That's why - without experimental proof - you cannot conclude anything outside the ranges where the formula is (with given certainty) valid.

At infinitesimal scale you end up with another problem - it is a quantum world. The term "surface area" as we understand it in our macroscopic world doesn't work anymore. Due to the probabilistic nature of the particle positions you cannot specify the area where those particles are (with 100% certainty). With the term "force" it's even worse. In quantum world you don't work with forces but potentials (force is related to potential as its' derivation). So you cannot just take the formula P = F / S and use it in a quantum world and expect it will work as-is.

Vladimir Fekete - 2 years, 9 months ago

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@Vladimir Fekete – exactly divide real world with micro and macroscopic world up to now, surface area is basis concept of this probem,I can not agree more. 'Break point' as you know, always inspire people to explore valid of theory boundary

丁大喵 by丁丁猫 - 2 years, 8 months ago

Consider it as a liquid bubble Surface tension causes pressure inside of a liquid bubble to increase within

Iqbal Mohammad - 2 years, 8 months ago

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