**Mathematical Procedure**

Consider the force diagram to the right, in which the deflection has been greatly exaggerated. The analysis has been simplified by considering the attraction on only one side of the mountain. A plumb-bob of mass `m` is situated a distance `d` from `P`, the centre of mass of a mountain of mass `M _{M}` and density

`ρ`. It is deflected through a small angle

_{M}`θ`due to its attraction

`F`towards

`P`and its weight

`W`directed towards the Earth. The vector sum of

`W`and

`F`results in a tension

`T`in the pendulum string. The Earth has a mass

`M`, radius

_{E}`r`and a density

_{E}`ρ`.

_{E}The two gravitational forces on the plumb-bob are given by Newton's law of gravitation:

Where `G` is Newton's gravitational constant. `G` and `m` can be eliminated by taking the ratio of `F` to `W`:

Where `V _{M}` and

`V`are the volumes of the mountain and the Earth. Under static equilibrium, the horizontal and vertical components of the string tension

_{E}`T`can be related to the gravitational forces and the deflection angle

`θ`:

Substituting for `T`:

Since `V _{E}`,

`V`,

_{M}`d`and

`r`are all known, and

_{E}`θ`and

`d`have been measured, then a value for the ratio

*ρ*:

_{E}*ρ*can be obtained:

_{M}Read more about this topic: Schiehallion Experiment

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