# The Naked Scientists Forum

### Author Topic: Why does the moon always show us the same face.  (Read 23588 times)

#### jysk

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##### Re: Why does the moon always show us the same face.
« Reply #25 on: 02/12/2006 14:57:18 »

This is because the gravitational field inside a uniform thick spherical shell is zero at all points inside the shell.

Is this what you really wanted to say?

I know this isn't a model we can recreate in a lab. Plus, the "Inverse/Squares Law" totally fails for resolving gravitation of two masses in contact, (denominator goes to Zero) but to make such a counter-intuitive statement seems against the spirit of this forum.

(Forgive my being so blunt.)

I think that as random points within the Earth are considered, the mass of material radiating out from all sides of each random point has an associated piece of "Acceleration Due To Gravity" to consider, one that is unique in every direction surrounding that point.

Arguably, there is one single point somewhere within the planet where g = 0, (And that point is not at the Earth's geometric centre) but since I've begun this sentence, that point of g = 0 has repositioned itself due to Tectonics.

My first question on the subject is; Can Newtonian mechanics be modified to reason out this model? Or is it more of a Quantum Mechanics model?

Mike
« Last Edit: 02/12/2006 15:12:46 by jysk »

#### daveshorts

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##### Re: Why does the moon always show us the same face.
« Reply #26 on: 02/12/2006 15:14:31 »
Quote
This is because the gravitational field inside a uniform thick spherical shell is zero at all points inside the shell.
This is actually a fairly standard result of newtonian gravity and vector calculus. I will see if I can work out how to derive, it and then if I can derive it in a comprehensible manner.

#### daveshorts

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##### Re: Why does the moon always show us the same face.
« Reply #27 on: 02/12/2006 16:15:35 »
Ok right here we go.

There is a theorem in mathematics called stokes theorem which like all the best maths is bloody obvious when you think about it but has profound consequences.

This says that for a substance that is conserved
Inside a closed surface - one that joins up like a sphere

The sum of the amount produced - the sum of the amount destroyed = the amount going through the surface.

If you think of the stuff as water this is saying, it has to go somewhere.

Gravitational field is a conservative field - it is conserved
and it is created by mass so we can rewrite the above as:

Sum of gravitational flux over a surface = Mass within this surface.

gravitational flux is just the field times the area you are looking at.

or in the terms of the above diagram - where g = the gravitational field at a point
ds = the bit of surface you ar looking at

sum the flux over the surface = sum of g x dS over the surface M

there is a constant in there we will call G the gravitational constant

s

Now if we consider a sphere everything is beautifully symmetric
so the field must be symmetric so g must be constant.

This is great because it means all we need to know to work out the gravitational field produced by a uniformly distributed mass in a sphere is it's Mass and the surface area

g Total mass/ Surface area

the area of a sphere is 4πr2 so:

g   M / 4πr2

Oh look if M is constant eg if we are outside the body we appear to have derived the inverse square law, which is reassuring.

This means that outside of a spherical body we can consider the field produced by it is identical to a point mass - which makes calculations far far easier.

Now if we consider a hollow sphere:

the field produced by the shell inside it inside must again be uniform:

Now the flux leaving outwards + the flux leaving inwards = Mass of the shell

We can work out the flux leaving outwards:

We worked out earler that a spherically symmetric body's field is the sum of it's symmetrically symmetric parts so:

flux leaving a solid sphere = flux leaving a sphere the size of the hole + affect of the shell

as the flux leaving the shell inwards will make no difference to the big shpere

fluxsolid = fluxhole + flux leaving shell outwards

this can be rearranged to:

flux leaving the shell outwards = fluxsolid - fluxhole

flux leaving the shell outwards Msolid - Mhole

We know that the
mass of the shell = mass of a solid sphere - mass of a sphere the size of the hole

So
Total flux leaving the shell mass of a solid sphere - mass of a sphere the size of the hole

Total flux leaving the shell = flux leaving the shell outwards

so there is none left to leave inwards so the field inside the hollow spherical shell must be zero!

I hope that makes some sense I apologise for using = & interchageably, if you did it all properly it would have the same result
« Last Edit: 16/12/2006 18:10:12 by chris »

#### syhprum

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##### Re: Why does the moon always show us the same face.
« Reply #28 on: 02/12/2006 18:11:11 »
Due to your elegant exposition I now understand how mass of residual part of the earth varying according with a third power and Newtons inverse square law combine to give a value of G that varies in a linear manner with distance from the centre.
Could you please now turn your attention to the transit time which I am surprised to learn depends solely on the density.
I am saddened to learn that my tunnel from the UK to the Antipodeon islands would not work due to the rotation of the earth it would have to run from North to South Pole which would involve rather a lot of air travel

#### daveshorts

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##### Re: Why does the moon always show us the same face.
« Reply #29 on: 02/12/2006 18:24:11 »
If the force varies linearly with distance from the centre of the earth this is the same relationship as force has with movement away from equilibrium in a spring.

A mass on a spring's period is constant as long as you don't change the spring or the mass. - it is a form of simple harmonic motion, you probably did it at school.

In this case changing the mass won't help you as the force is proportional to the mass. Changing the size of the planet is just like changing the starting amplitude of a mass on a spring. So the only thing you can change which will have an effect is the density of the planet, or of course G.

#### Titanscape

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##### Re: Why does the moon always show us the same face.
« Reply #30 on: 03/12/2006 15:29:20 »
As I see it, the earth was struck, then a small planet, by a Mars sized object, or so I heard. Then a mass was sent out that had been part of the earth and was in motion with it and so it left with a similar motion and hence rotates facing the earth, slowly turning. It spun off, the earth didn't quite divide in two. All the solar system rotates on the same plane and the smaller planes of the sattellites are aligned too.

All from one disc which was originaly all dust from another star which exploded. Then gravity caused the formation of a centre, the sun, and then planets and then sattelites and comets and there were a lot of collisions. Look at the moon. Venus was knocked backwards and Uranus lost it's axis. Pluto was detached from it's planet.

The earth's rotation must have changed by this impact and over time it slowed down because of the seas and the moon tides.
« Last Edit: 03/12/2006 15:31:21 by Titanscape »

#### jysk

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##### Re: Why does the moon always show us the same face.
« Reply #31 on: 05/12/2006 04:40:13 »
Thank you for your explanation Daveshorts

Mike

#### Soul Surfer

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##### Re: Why does the moon always show us the same face.
« Reply #32 on: 06/12/2006 10:47:28 »
The linear fall off is a bit counter intuitive, but it is true it results in the motion of the body through the center of the earth being a classical simple harmonic motion.  In fact you can think of it as a completely flattened orbit  if you looked at a circular orbit going round the earth from the side a linear "orbit" through the middle would look just the same.

#### syhprum

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##### Re: Why does the moon always show us the same face.
« Reply #33 on: 06/12/2006 11:33:11 »
Many points of this strange business become progressively more clear the path of a straight down and up trip is of course elliptical but with one axis reduced to zero, I would still like to calculate the transit time on the basis of how G varies as you go down and up if you have any idea how it could be done?

#### daveshorts

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##### Re: Why does the moon always show us the same face.
« Reply #34 on: 06/12/2006 11:39:59 »
Do you want to work out the transit time for any variation of g or for a linear change. If the first have you done any calculus?

#### lyner

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##### Re: Why does the moon always show us the same face.
« Reply #35 on: 06/12/2006 12:10:38 »
The Hole Through the Earth Problem with some practicalities and simplifications.

This is an interesting thought experiment and involves being dropped through a hole drilled between the North and South Poles. (If you take any other journey, you would bash into the sides as the Earth rotated.)

Let's assume, for a minute, that the hole is completely empty and ignoring the incredibly high temperatures and pressures down there (probably 5000 degrees and about 5 million times atmospheric pressure).

We start to fall down the hole; accelerating under 'normal gravity'.  Newton tells us that only the bits of the Earth that are nearer to the centre than we are would have any effect on us. The bits nearer the surface than we are have no net effect; their attractions would all cancel out.

This force on us is, in fact, proportional to our distance from the centre. When half way down, we would weigh half as much as at the surface. This means that we exhibit what is known as Simple Harmonic Motion (just like a pendulum or a weight on a spring). We would oscillate backwards and forwards, from pole to pole for ever, if it weren't for friction.

The period of this motion is fairly easy to calculate. It's about 90 minutes for the return journey; exactly the same as the time for a satellite in a low Earth orbit to go round the Earth once.

To be a bit practical; what would be the effect of the hole being open to the air? Well, we know that, as we go up in the air, the atmospheric pressure halves every 5.5km. As we went down the hole, the reverse would happen; every 5.5 km downwards, the pressure would double. After only 110km (only 2% of the way down) the pressure would have doubled 20 times! That corresponds to a million times atmospheric pressure. The air molecules would have long since been squeezed together and this simple calculation runs out of steam. But, in any case, the atmosphere would become a highly compressed liquid and we would never get through,

We could do this experiment though, if we went out into space and found ourselves a nice, spherical asteroid or tiny planetoid of the same density as the Earth - say it was a couple of hundred km across and made of Granite If our planetoid were in an orbit near to Earth, it would be warm enough so that any atmosphere would have boiled away so the hole would be completely empty all the way through.

It would work.

But here is the interesting bit. Because we are dealing with simple harmonic motion, the return trip would still take 90 minutes and this would also be the orbit time of a tiny satellite in orbit just above its surface. It must also be true for bits of dust orbiting close to rocks. I wonder if astronauts have ever observed it outside their window. .

#### syhprum

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##### Re: Why does the moon always show us the same face.
« Reply #36 on: 06/12/2006 15:40:03 »
Do you want to work out the transit time for any variation of g or for a linear change. If the first have you done any calculus?
I had about three months of calculus lessons in 1944 but we were only spoonfed various formulae with lille idea of practical applications.
I wanted to calculate the the time of descent to the centre allowing for the fact that G was decreasing in a linear manner in accordance with how far we have travelled.
If I know G and either the time or the distance I can easily calculate the other but when two things are varying simultaneously I am defeated

#### Atomic-S

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##### Re: Why does the moon always show us the same face.
« Reply #37 on: 11/12/2006 06:32:59 »
g = GM/r^2  G being the universal gravitational constant and M the mass of the
portion of the earth which is active, which is that contained inside a sphere having a radius equal to the distance from the center to the traveler (assumed inside the earth).  Thus, the force upon the traveler is proportional from the distance to the center, and what you have here is a simple harmonic oscillator, like a mass connected to a linear spring.

#### syhprum

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##### Re: Why does the moon always show us the same face.
« Reply #38 on: 11/12/2006 09:33:42 »
It is simple to calculate the down-up time on the basis that it equal to the orbital time, what I was hoping was that someone with the necessary mathematical skills would show how it can be calculated in an alternative manner by allowing how the velocity of the elevator varies under the influence of the steadily decreasing gravitational force as it travels towards the centre of the Earth

#### lyner

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##### Re: Why does the moon always show us the same face.
« Reply #39 on: 16/12/2006 14:45:02 »
To calculate it you just need to do the same bit of bookwork that you use for normal simple harmonic motion. You solve the second order differential equation which gives you a sinusoidal answer for the  motion.
Of course, it is only sinusoidal for a  sphere of uniform density. Any random old planet would need the calculation to be done step by step, using numerical methods (a fast computer) - but then what would that prove unless you actually NEEDED to know the answer in detail? Calculus is much more satisfying.
lyner

#### MoBiJo

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##### Why does the moon always show us the same face.
« Reply #40 on: 12/02/2007 09:54:52 »
I have visited Australia several times and have noticed that the water spirals down the waste pipe the the same way relative to the earths rotation as it doe's in Britain.
The water does not rotate in a different direction it is just that you are viewing it from a different perspective

"viewing it from a different perspective" doesn't makes sense. clockwise is still clockwise and counterclockwise is still counterclockwise, even on the southern hemisphere ;)

#### MoBiJo

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##### Why does the moon always show us the same face.
« Reply #41 on: 12/02/2007 10:05:05 »
Thank you daveshorts and Soul surfer,

your two explanations have completly satisfied me as to the original question, why we always see the same face of the moon.
« Last Edit: 12/02/2007 10:07:06 by MoBiJo »

#### The Naked Scientists Forum

##### Why does the moon always show us the same face.
« Reply #41 on: 12/02/2007 10:05:05 »