Q3.11 G and g (S)

What is the relationship between G, the universal constant of gravitation, and g, the acceleration due to gravity at the earth’s surface? Appeal to this relationship to find a value for g.
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Hint

Think of a body of mass m at the surface of the earth. There are two ways of expressing its ‘weight’ (the gravitational force exerted on it by the earth). The first involves g; the second involves G. They have to match up! You will also have to look up the values of some physical constants.
 

Solution

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Solution

Consider a body of mass m at (or near) the surface of the earth.

The gravitational force $\vec{F_G}$ the mass experiences due to the earth (its weight) can be written in either of two ways.

First we can appeal to the usual expression for the (magnitude of) the weight:

FG=mg
Second, we can appeal to the Law of Gravitation
\[ F_G=\frac{GM_Em}{R_E^2} \]
Here ME is the earth mass, while RE is the radius of the earth, and also, therefore the separation between the mass m and the centre of the earth. Recall that it is legitimate to treat the earth as a point mass, with all the mass concentrated at its centre.

Since the two expressions must agree we can equate them:

\[ mg = \frac{GM_Em}{R_E^2} \]
so that
\[ g = \frac{GM_E}{R_E^2} \]
Substituting G=6.67×10-11Nkg-2m2, RE=6.37×106 m and ME=5.97×1024 kg gives g=9.81 ms-2. The measured value is close to this; reasons for departures from this value include the earth’s rotation and the fact that the earth is not a perfect, uniform sphere. Geologists might be interested in noting how variations in g reflect local geological structure .