Fun Q&A: Gravity of the Earth - Answers Surprising!

[Andre]

Just for some fun.

Q1 Where is gravity of the Earth the strongest?

A - On the surface of the Earth
B - In the centre of the Earth
C – Somewhere about halfway down to the centre of the Earth


Q2 If you dig yourself about one thousand of miles into the Earth, what happens to gravity?

A. - Increases definitely.
B. - Stays more or less constant.
C. - Decreases definitely.

There is more to it than the eye meets.

 

[jcsd]

Q1 on the Earth's surface

Q2 C

 

[Andre]

The logical answer. But perhaps think again. Where is the error in the reasoning? It's definitely a bit more complicated.

 

[Andre]

OK here is a similar discussion that leads to the obvious answers Q1 - gravity of the Earth the strongest on the surface of the Earth and Q2 - gravity decreases definitely with digging a thousand miles into the Earth.

However, this is the complication.
In particular this.

Now using those data in a basic numerical intergration spreadsheet gives a surprising result.

Guess again.

 

[Njorl]

I don't have any desire to do the math, but you'd do it like this:

Assume a set of spherical shells of differing density. If you want to be really accurate, assume density is a dependent variable of r, distance from core center. Integrate, and use Gauss' law to get gravity as a function of radius. Differentiate to find minima and maxima (there will also be singularities due to discontinuities in the density), explore each to see if it is a local or general maxima.

I would guess that gravity will be highest at some transition from a very dense to non-dense material. The starkest of these is the transition from solid to gaseous - the surface on which we live, but it is not the only one.

Njorl

 

[Andre]

Well the way I figured:

For a homogenouos sphere it can be determined that g(r)=GM(r)/r2. M(r) being the mass within of the sphere with radius r. - G gravitational constant and rho density (Sorry, not proficient with that excellent formula tool)

The total mass outside that radius is cancelling each other. Since M(r)=4/3 * rho * pi *r^3 it follows that g(r)=4/3 * G * rho * pi * r. In other words the gravity inside a homogenouos sphere is directly proportional to the distance to the centre.

Now assuming that the density rho is linear decreasing from value a with increment b, we replace rho with rho(r)=a-br. Substitute it in the first formula to get only a simple square formula, so intergrating is basic. Can be done on the stamp of that old envellope

However we have five discontinuing layers: inner core, outer core, lower mantle, upper mantle, lithosphere. Each with their own numbers r-min, r-max, rho-a and b

So a practical approach is numerical intergration in a spreadsheet with the advantage having the graph visible immediately. I used 100 km increments for each shell. Cross checking it with the real Earth data (mass and g) the numbers in that link appeared to be off by 1,6%.

Now who wants to predict what my graph shows?

 

[Andre]

Ok, http://home.wanadoo.nl/bijkerk/gravity.jpg

First of all, in my previous post I mentioned an error of 1,6% with real Earth data. However, I discovered that at that moment the crust/lithosphere data were overwritten by mantle data. When I corrected that, the error increased to 2.2%. So I increased all the density data with a factor 1.022 to bring the gravity error and total Earth mass error with 0,1%. I wonder what is wrong? The approximation with increments was from the top side, so the result should have been higher than the reality.

Now, the answers are clear albeit a bit surprising. Gravity remains more or less constant initially. Why? We still can neglect the mass outside the radius when heading for the centre - this is decreasing the gravity. But we also come closer and closer to the very dense core - this is increasing the gravity. Both are of about the same magnitude initially, cancelling each other. (note that in the initial stage, entering the (relatively very light) crust the gravity increases sharply from 9.81 to 9.96 ms-2)

Coming closer to the the core however, the increasing factor wins and my model indicates a maximum value of 10,62 ms-2 at the core mantle boundary versus 9,81 at the Earth surface. Njorl predicted that more or less correctly.

Inside the cores the behaviour is more or less approaching the homogeneous sphere - lineair proportional to the radius. Now isns't that a nice to know for discussions. So concluding:

Q1 Where is gravity of the Earth the strongest?

A - On the surface of the Earth
B - In the centre of the Earth
x C – SOMEWHERE ABOUT HALFWAY DOWN TO THE CENTRE OF THE EARTH. 10,62 ms-2 at a dept of about 2890 km


Q2 If you dig yourself about one thousand of miles into the Earth, what happens to gravity?

A. - Increases definitely.
X B. - STAYS MORE OR LESS CONSTANT. (fluctuates actually between 9.96 and 10.04 ms-2)
C. - Decreases definitely.

There is more to it than the eye meets

Actually the whole exercise was meant initially to calculate the relative order of magnitude of the angular inertia of the cores and the mantle (to get the angular momentum). These calculation are very similar. The result is that the inner core contributes 0,35% to the angular momentum of the Earth, the outer core 14,22% and the mantle/crust 85,43%. Anybody recognises those numbers? I infer that the angular momentum of the core is sufficient to exert influence on the mantle in case of anomalies, in contrast to a statement of a geophysisist during a discussion.