Everything about E Tv S Effect totally explained
In the early 1900s a German team from the Institute of
Geodesy in Potsdam carried out gravity measurements on moving ships in the Atlantic, Indian and Pacific Oceans. While studying their results the Hungarian nobleman and physicist
Roland Eötvös (1848-1919) noticed that the readings were lower when the boat moved eastwards, higher when it moved westward. He identified this as primarily a consequence of the rotation of the earth. In 1908 new measurements were made in the Black Sea on two ships, one moving eastward and one westward. The results substantiated Eötvös' claim. Since then geodesists use the following formula to correct for velocity relative to the Earth during a measurement run.
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The second term represents the required
centripetal acceleration for the airship to follow the curvature of the earth. It is independent of both the earth's rotation and the direction of motion. For example, when an aeroplane carrying gravimetric reading instruments cruises over one of the poles at constant altitude, the aeroplane's trajectory follows the curvature of the earth. The first term in the formula is zero then, due to the cosine of the angle being zero, and the second term then represents the centripetal acceleration to follow the curvature of the Earth's surface.
Explanation of the cosine in the first term
The mathematical derivation for the Eötvös effect for motion along the Equator explains the factor 2 in the first term of the Eötvös correction formula. What remains to be explained is the cosine factor.
Because of its rotation, the Earth isn't spherical in shape, there's an
Equatorial bulge. The force of gravity is directed towards the center of the Earth. The
normal force is perpendicular to the local surface.
On the poles and on the equator the force of gravity and the normal force are exactly in opposite direction. At every other latitude the two are not exactly opposite, so there's a resultant force, that acts towards the Earth's axis. At every latitude there's precisely the amount of centripetal force that's necessary to maintain an even thickness of the atmospheric layer. (The solid Earth is ductile. Whenever the shape of the solid Earth isn't entirely in equilibrium with its rate of rotation, then the
shear stress deforms the solid Earth over a period of millions of years until the shear stress is resolved.)
Again the example of an airship is convenient for discussing the forces that are at work. When the airship has a velocity relative to the Earth in latitudinal direction then the weight of the airship isn't the same as when the airship is stationary with respect to the Earth.
If an airship has an eastward velocity, then the airship is in a sense "speeding". The situation is comparable to a racecar on a banked circuit with an extremely slippery road surface. If the racecar is going too fast then the car will drift wide. For an airship in flight that means a reduction of the weight, compared to the weight when stationary with respect to the Earth.
If the airship has a westward velocity then the situation is like that of a racecar on a banked circuit going too slow: on a slippery surface the car will slump down. For an airship that means an increase of the weight.
The first term of the Eötvös effect is proportional to the component of the required centripetal force perpendicular to the local Earth surface, and is thus described by a cosine law: the closer to the Equator, the stronger the effect.
Motion along 60 degrees latitude
The same gravimeter is used again, its internal weight has a mass of 10,000 grams.
Calculating the weight reduction when stationary with respect to the Earth:
An object located at 60 degrees latitude, co-moving with the Earth, is following a circular trajectory, with a radius of about 3190 kilometer, and a velocity of about 233 m/s. That circular trajectory requires a centripetal force of about 0.017 newton for every kilogram of mass; 0.17 newtons for the 10,000 gram internal weight. At 60 degrees latitude, the component that's perpendicular to the local surface (the local vertical) is half the total force. Hence, at 60 degrees latitude, any object co-moving with the Earth has its weight reduced by about 0.08 percent, thanks to the Earth's rotation.
Calculating the Eötvös effect:
When the airship is cruising at 25 m/s towards the east the total velocity becomes 233 + 25 = 258 m/s, which requires a centripetal force of about 0.208 newtons; local vertical component about 0.104 newtons. Cruising at 25 m/s towards the west the total velocity becomes 233 - 25 = 208 m/s, which requires a centripetal force of about 0.135 newtons; local vertical component about 0.68 newtons. Hence at 60 degrees latitude the difference before and after the U-turn of the 10,000 gram internal weight is a difference of 4 gram in measured weight.
The diagrams also show the component in the direction parallel to the local surface. In
meteorology and in
oceanography, it's customary to refer to the effects of the component parallel to the local surface as the
Coriolis effect.
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