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Earth`s Gravitation

Isaac Newton was the first to explain the motion of heavenly bodies by employing the concept of gravitation. Newton came up with the concept of gravity after he studied the motion of the Moon and the planets for many years.

If m1 and m2 are the masses of two point bodies and r is the distance between them, then the gravity law is expressed as
F = Gm1m2
Where G = 6,67∙10-11 N∙m2/kg2 is the gravitation constant. This law also works for the spherically symmetric bodies (when the distances between the bodies' centers are longer than the sum of their radiuses). The law of gravity can be implemented for calculating gravitational relationships between any bodies, provided that the distance between them is significant as compared to their sizes.

Acceleration of the body m at distance r from the body M is equal to a = F/m = GM / r2.

Gravitational acceleration in the Earth's field is equal to GM / r2, where M = 5,97∙1024 kg is the Earth's mass, and r is the distance to its center. Gravitational acceleration near the Earth's surface is equal to g = 9,8 m/s2. The Earth's oblate shape and rotation result in the difference of gravitational forces at the equator and near the Earth's poles. Gravity acceleration at any given observation point may be approximately calculated by formula g = 9,78 ∙ (1 + 0,0053 sin φ), where φ is the width of this point.

If we consider the Earth as a perfect sphere, then the gravity force within it weakens pro rata to distance from its center in accordance with:
g(r) = G M r
The left side of the model displays the Earth with a tunnel at distance L from its center. This distance may be changed. The bottom right angle displays the diagram of gravitational acceleration dependence on the distance to the Earth's center. The gray line outlines the Earth's radius, which is equal to 6,371 km.

Press "Run" button to observe a body falling into the shaft. (Note that this model does not account for the Earth's frictional and rotational forces. The only force that acts upon the body as it falls is the force of gravity.) Having passed through the Earth's center at maximal speed, the body's fall starts to slow down. The body stops completely upon reaching the opposite end of the tunnel, and starts falling back. In this way, it goes through a kind of an oscillating motion. The diagram of the body's acceleration over time is shown in the model's top right corner.

Body motion may be suspended or stopped by pressing "Stop" and "Reset" buttons, respectively.

© OpenTeach Software, 2007