Gravitational Force

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Newton’s law of universal gravitation states that any two elements of mass in the universe attract each other with a force proportional to their masses and inversely proportional to the square of the distance separating them. Thus, if two mass elements M and m are separated by a distance r ≡ | Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \vec{r} | (with the vector | Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \vec{r} | directed toward m, then the force exerted by mass M on mass m due to gravitation is:


Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): g(r) = -\frac{GMm}{r^2} (\frac{\vec{r}}{r})


where G is a universal constant called the gravitational constant. Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): G = 6.67×10−11 N·(m/kg)2

For meteorological purposes, we assume that the mass of the Earth, M, is much larger than the point mass, m.

Additionally, we assume that the distance between the point mass and the center of the Earth is approximately the radius of the Earth, 6371 km. We can make this assumption because the troposphere (where ~80% of the atmosphere's mass is) is only about 10km thick.

With these assumptions, the equation simplifies to:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): g(r) = -\frac{GM}{r^2} (\frac{\vec{r}}{r})

We normally assume this value of the gravitational acceleration to be constant at Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 9.80665 m/{s^2}