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No 63 - July 2010

Insights into gravitation on Earth

Last November, I described the calculations of the gravitational attraction to various bodies that are not spheres. The results were written as tables and diagrams, as no formulae can be derived for the complicated integrations involved. The sphere also needs a significant integration but by contrast, there is only one answer, the distance to the centre of the sphere; this is then used with Newton's third law to generate the gravitational pull.

However if one poses the question "What on earth affects our weight most?" the simple spherical form allows one to find out what affects us most (aside from the frivolous answer of "How much we eat!").

gravity - full earth gravity - half earth
Fig. 1. Fig. 2.
The unit of length is the earth's radius (about 4000  miles). The units of mass and gravity g are also the earth's mass and gravity. Gravity is then mass divided by the square of the radius g = M / R2 and the distance to the gravitational mass centre is R = √(g/M).

This shows the earth with normalised units. One cubic metre at the centre weighs comparatively little, but if it lies just 4 miles below your feet, it would have 1 million times more pull via the inverse square law. As 4 miles is about 6 km, there are another 12,000 half-metres to go before one is physically standing on it. This would involve multiplying by a further 144 million times!

This gives the impression that most of the pull comes from the vicinity of your feet! However, this cannot be true, or one would find oneself leaning sideways when next to a cliff. (In fact, one really is, but it can hardly be measured at all with the most sensitive instruments.) A better insight is obtained by doing a calculation on a smaller part of the earth.

This shows that half our weight comes from a sphere that just reaches the centre. As this is only one eighth of the earth's volume, it is apparent that the remaining 7/8 is rather ineffective in supplying the other half. This is made obvious by the addition of a second half-radius sphere below.

gravity - two half earths in line gravity - two half earths transverse
Fig. 3. Fig. 4.

As its centre is three times further away than the upper sphere, it exerts one-ninth of its g, giving only a slight increase to 0.5555. As the total weight has now doubled, the centre of attraction - given by inserting the known mass and g into the inverse square law - moves noticeably down from the centre of the top sphere, to 0.67. This is exactly the same figure given for the calculation in Journal 61 of a split mass in line with the observer. In that case, the mass size was not defined; indeed, it has no effect on the calculation as long as it remains a sphere.

Such a body when rotated through 90 degrees is only slightly more difficult to calculate.

In addition to the downward displacement of the centres, one has to allow for the lengthening of the hypotenuse and the cosine reduction due the angle of the vector away from the vertical. The result is to move the gravity mass-centre noticeably below the line joining the centres.

Finally, a sphere approaching the full size of the earth gives some interesting results.

gravity - 90% earth gravity - excavated 90% earth
Fig. 5. Fig. 6.
√[(1 - r3) / (1 - r)] = √(1 + r + r2)

72% of the full earth's mass is sufficient to produce 0.9 g and the remaining 28% would produce the other 10%. The mass still operates from its centre, but consider the "excavated" situation of Fig. 6.

This now operates from much further down, at 1.65 earth radius. It does not, however, tend to the centre of gravity, even when the excavation is increased to the full earth. From the formula in the figure it can be seen that 1-r (for g) divides into 1-r3 for the mass, to give 1+r+r2. As r tends to unity this goes to 3, so the mass centre never goes below √3 in this "wine glass" shape. So one is not attracted to the "wine", but to a point in the "glass" 1.73 radii below the top. :-)

This feature arises from the fact that though there is decreasing mass in the upper sides; the inverse square law there is exerting its strong influence, even allowing for the cosine reduction. This is unlike the situation where the full body is distanced from the measuring point.

Graham Clarke


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