# Tricky question

1. Jun 25, 2004

### 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.

Last edited: Jun 25, 2004
2. Jun 25, 2004

### jcsd

Q1 on the Earth's surface

Q2 C

3. Jun 25, 2004

### Andre

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

4. Jun 25, 2004

### 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.

5. Jun 25, 2004

### 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

6. Jun 25, 2004

### 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?

Last edited: Jun 25, 2004
7. Jun 26, 2004