[Physics] Will the sea form a concave surface


Without neglecting the fact that the earth is rotating, let us be in a ship (such that there is no relative motion between sea and ship) in a sea that surrounds the northern most or southern most point of earth's rotational axis.

Will the ship crew see a concave surface water (using Newton's rotating bucket argument)?

rotating bucket
(source: books24x7.com)

rotation of earth at poles
(source: physicalgeography.net)

Best Answer

No, it won't.

In a bucket, the gravitational forces acting on each particle of the fluid are parallel (if the the bucket is small with respect to the Earth radius, which I think we can always assume). Without rotations, the fluid will form a plane surface inside the bucket (which is neither convex nor concave). This surface coincide with a surface of constant potential energy (given by the gravitational field of the Earth). In the presence of a centripetal force, the surface of constant potential energy is now a concave surface, due to the presence of a potential energy term given by the centripetal field.

The gravitational forces acting on each particle of the ocean's water are not parallel (in fact the Earth is a sphere). The centripetal force due to Earth rotation will not make the ocean concave. The effect of centripetal forces in this case is the fact that the Earth radius is larger at the equator than at the poles. As a consequence of this, the shape of the Earth is that of an ellipsoid of revolution. To see this more clearly, one can calculate the potential energy of a point on the surface, considering the sum of gravitational acceleration and centripetal forces. The sea level coincide with a surface of constant potential energy.

Note that centripetal force do not only deform the shape of the waters from a perfect sphere to an ellipsoid, but they deform also the Earth crust.

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