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Gravitational Fields 1


Newton's Law of gravitation

variation of g with distance




Newton's Law of Gravitation



For two masses displaced a distance apart, the gravitational force of attraction of one


mass on the other is directly proportional to the product of the two masses and


inversely proportional to the square of the distance between them.




Newton's Law of Gravitation



If the masses are m1 and m2 , with their centres of mass displaced a distance r apart, then the force of attraction F of one mass on the other is described as:


newton's law of gravity


The proportionality can be made into an equation using a constant of proportionality. This constant we call G, the Universal Gravitational Constant.


equation #2


G = 6.67 x 10-11 N m2 kg-2



Gravitational force is very weak! This can be shown by considering two 1 kg masses 1 m apart. The gravitational force between them is given by:


force between two 1kg masses 1m apart


The gravitational force between everyday objects is so small as to be almost irrelevant.



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Variation of 'g' with distance from the Earth's centre


To understand this work we must recall that 'g' is the acceleration due to gravity (9.8 ms -2). This value is for the surface of the Earth.


Weight W is the force of attraction of the earth on a mass.


For a mass m, the weight is given by:


weight in terms of mass - equation #5



The mathematical treatment depends on two assumptions:


1. The value of g is the same at a distance from a mass, whether the mass is in the shape of a spherical shell or concentrated in the centre.



g the same whether sphere or shell




2. The value of g everywhere inside a spherical shell is zero.


NB The spherical shell and central mass have uniform density.



First consider a mass m on the surface of the Earth. The force of attraction between the mass and the earth is its weight W. This is also equal to the force F between the mass and the Earth, given by Newton's Law.


weight at the surface of the Earth - equation #6


        weight at the surface of the Earth - equation #7


g at a distance r - equation #8                (i             


where ME is the mass of the Earth, rE its radius.



Now let us consider the value of g at a distance r from the Earth.



case where r > rE


If this new value is gr , then by similarity with equation (i,


g at a distance r - equation #9                 (ii         


Dividing equation (ii by equation (i,


g at distance r - equation #10(iii             


case where r < rE



g inside the earth



In the diagram the point X is inside the earth at a distance r from the centre.


From our initial assumptions, the value of gr is a result of the gravity from a sphere of radius r .

If MS is the mass of the sphere, then by comparison with equation (i ,


g at a distance r for a hollow sphere


mass of sphere equation #15                 (iv                 


NB the effect of matter (in the form of a shell) above point X has no effect on the value of gr

let us assume that masses have uniform density ρ (rho).

Remembering that m = ρV , the mass MS of the internal sphere and the mass ME of the Earth is given by:


mass of a sphere of radius r - equation #12


mass of the earth in terms of its radius -equation #13


Dividing the first equation by the second,


mass of the earth - equation #14


Substituting for MS from equation (iv ,


equation #16


equation #17


recalling that equation #8


equation #18




g vs r graph




NB   gr = g when r = rE





So for inside the Earth, gr is directly proportional to r . The graph is therefore a straight line through the origin.


For outside the Earth, gr follows a function similar to y = x-2 , where x decreases steadily, approaching zero at infinity.


equation #20


where rE and g are constants





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