Newton’s Law of Gravitation

We all feel the pull or attractive force of the Earth’s gravity, particularly when we lift a heavy mass. This pull is due to the attraction of all the material of the Earth but depends most on the rocks below where we experience the pull. Therefore, by measuring how the pull varies from place to place we can make deductions about the subsurface. Evidently, we need to understand how subsurface rocks – and other fac- tors – affect the pull of gravity at the Earth’s surface. Most people know the story about Sir Isaac Newton discovering gravitation after being hit by a falling apple. Of course, people did not need New- ton to tell them that apples and other objects fall; what Newton had the genius to realise is that the familiar force that pulls objects towards the ground is also responsible for holding planets in their orbits around the Sun, and in fact that all bodies attract one another. He found that the attractive force (Fig. 8.1) between two small masses, m₁ and m₂, is pro- portional to the product of their masses (e.g., dou- bling either mass doubles the force, doubling both quadruples the force) and inversely proportional to the square of their separation, r (doubling their sep- aration reduces the attraction to a quarter). These relationships are summed up by the equation

Eq. 8.1

attractive force, F Gm m Newton’s Law

r = 1 2 2 107 force force m₂ m₁ r

G, the universal gravitational constant, (called ‘big gee’) has the value 6.672 × 10–8_m3_{/Mg · s}2_when

m₁ and m₂ are measured in Mg (tonnes), and r in metres. (Mass is often given in kg; Mg has been adopted here because then the density of water is 1, which is helpful for comparison; see Section 8.2.). The force between two everyday objects, such as people and vehicles, is extremely small, so we are not aware of it; only because the Earth is so large is its pull noticeable.

Newton’s Law applies in the first place to ‘point’ masses, that is, bodies so small in extent compared to their separation that all parts of one mass are closely the same distance from the other. A body that is extended is treated as an assembly of many small masses: The force on each component mass is calculated and these are added together; Figure 8.2 shows the pulls of such two small masses, i and j. This calculation can be laborious (especially as the forces are not parallel), but some shapes are fairly easy to evaluate. The attraction of a uniform shell – a thin, hollow sphere – turns out to be exactly the same as if all its mass were concentrated or shrunk to its centre (Fig. 8.3) (this is true only outside the shell; for instance, at its centre, the force must clearly be zero, because of the symmetry, as for every small mass pulling in one direction there is another pulling in the opposite direction).

This simplifies the calculation for any body con- sisting of concentric layers, which we know from seismology is approximately true for the Earth (Sec- tion 4.3). Each layer can be shrunk to a central point mass, and therefore the whole Earth behaves – so far as its external gravitational attraction is con- cerned – as if all its mass ME were concentrated at

the centre. Thus the Earth’s attraction for a small mass, m_S, at its surface is the same as if there were two point masses separated by the Earth’s radius, R_E (Fig. 8.4). This is essentially the same as Figure 8.1, so Eq. 8.1 can be used:

Eq. 8.2

Measuring the Earth’s pull by the force on a sur- face mass is not convenient because it depends on the mass mSas well as on the Earth; instead, we use

the acceleration that the force produces. If m_S is released – like Newton’s apple – just above the sur- face of the Earth, the force F causes it to fall with an acceleration g. As force = mass × acceleration,

Eq. 8.3

Eq. 8.4

m2cancels from both sides of the equation because

all masses fall with the same acceleration (provided they are dropped in a vacuum, which eliminates air resistance). Here g (called ‘little gee’ to distinguish it from G) is the acceleration due to gravity. It can be measured by letting a mass fall and timing how fast it accelerates or in other ways, as we shall see later. g GM R E E = ₂ F m g GM m R S E S E = = ₂ force of attraction, F GM m R E S E = ₂ m2 rj ri i j

Figure 8.2 Gravitational pull of an extended body.

m_S M_E

RE mS

R_E

=

Figure 8.4 The Earth approximates to a point mass.

m2 shell

To the extent that the Earth approximates to spherical symmetry, as shown in Section 4.3, the value of g is the same everywhere on the surface and would be the same no matter how its mass is distrib- uted internally, provided it is spherically symmetrical. Therefore, the Earth might – so far as its surface gravity reveals – have a hollow centre; alternatively, it might consist of pumice in the top several thousand kilometres with a sphere of gold within it, provided the different models have the same total mass. How- ever, we know the Earth is not quite spherically sym- metrical, if only because rock type varies laterally, so we expect there to be small variations of g over its surface. The size of these variations depends upon the extent of subsurface bodies and on how much their densities differ from those of surrounding rocks.

8.1.1 The mass of the Earth

The mass of the Earth can be calculated using Eq. 8.4. The value of G is known (by measuring, in a laboratory, the tiny force between two masses), g is found by timing the acceleration of a dropped mass, and the radius of the Earth can be found by survey- ing, leaving only M_Eto be deduced.

M_E is about 5.97 × 1021_{Mg (10}21_{is 1 followed}

by 21 zeros), or 5.97 × 1024_{kg. This is not a readily}

graspable number, but it is equivalent to the Earth having an average density of 5.5 Mg/m3_{, or 5}1_⁄2 times that of water. However, the densities of com- mon surface rocks, such as sandstone, limestone, granite, and basalt, are much less, seldom exceeding 3 Mg/m3 _{(see Table 8.1). This tells us that the}

Earth’s interior is made of denser material. This is due in part to the compression by the overlying lay- ers and in part to the interior being composed of more dense materials; for example, studies of mete- orites believed to come from a disrupted planet sug- gest that the core is mostly iron. (The deduction of the internal composition of the Earth is explained in, e.g., Brown and Mussett, 1993).