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# Why does the value of 'g' decrease as we go into the Earth?

Question #52306. Asked by shawn888. (Nov 10 04 8:22 AM)

Baloo55th

I thought we'd dealt with this one in the question about the tunnel through the earth. Opinion was that when you were at the centre, the earth would be pulling at you equally in all directions. So as you go down from the surface, there is an increasing amount of earth behind you, with a tendency to pull you back up. This cancels out some of the greater pull downwards from the rest of the earth below you. (Simplified, OK, you exert a pull on the earth, etc...)

 Nov 10 04, 8:36 AM
peasypod

OK here is a sketch of a proof.

The gravitational force is given by F = GmM/r^2
(r^2 denotes "r squared)
Where G is the universal gravitational constant
m is the object mass
M is the planet mass
r is the distance from the centre of the planet.

Since F = mg, where g is the graviational acceleration this gives
g = GM/r^2
whish is independent of the object mass

However, this relation, where M is the total planet mass is only true if you are outside the planet's radius. If you dig a hole, you approach the centre more closely, but now there is a shell of mass that should be ignored (the sum of all the forces from this shell will always be zero).

The mass now inside the radius falls off with a factor (r/R)^3 since it is determined by the relative volume described by your distance from the planet centre r, and the planet radius R.

This gives a relation for the gravitational acceleration at a general radius r < R.

g = GMr/R^3

R, M and G are constants....so g increases with r, and hece decreases as you approach the centre of the earth. (for the case r > R the first formula applies, which drops off with r^2, so the graviational acceleration is a maximum at the surface of the earth).

Of course all this assumes that the density of the planet is a constant, which the earth is to a first approximation.

OK Baloo...how deep a hole have I dug for myself here?

[Nov 10 04 4:16 PM] peasypod writes:

From memory the values of G, R and M for the earth are given by...
M = 5.98 x 10^24 kg
G = 6.67 x 10^-11 (units fail me, but its SI)
R = 6.4 x 10^6 m
Just in case you are interested in verifying that the results above hold water....

 Nov 10 04, 4:13 PM

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