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Question
The free surface of a liquid resting in an inertial frame is horizontal. Does the normal to the free surface pass through the centre of the earth? Think separately if the liquid is (a) at the equator (b) at a pole (c) somewhere else.
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Solution
Yes, the normal to the free surface of the liquid passes through the centre of the Earth. Because of the gravitational force, the free surface of the liquid takes the shape of the surface of the Earth. Also, because the gravitational force is directed towards the centre of the Earth, the normal to the free surface also passes through the centre of the Earth (in all cases).
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