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For astronomical bodies other than Earth, and for short distances of fall at other than "ground" level, g in the above equations may be replaced by (+) where G is the gravitational constant, M is the mass of the astronomical body, m is the mass of the falling body, and r is the radius from the falling object to the center of the astronomical body.
The mass of any of the discs is the mass of the sphere multiplied by the ratio of the volume of an infinitely thin disc divided by the volume of a sphere (with constant radius ). The volume of an infinitely thin disc is π R 2 d x {\displaystyle \pi R^{2}\,dx} , or π ( a 2 − x 2 ) d x {\textstyle \pi \left(a^{2}-x^{2}\right)dx} .
General triangular prism: ... r = the radius of the cone's base h = the distance is from base to the apex ... Solid sphere: r = the radius of the sphere ...
The theorem tells us how different parts of the mass distribution affect the gravitational force measured at a point located a distance r 0 from the center of the mass distribution: [13] The portion of the mass that is located at radii r < r 0 causes the same force at the radius r 0 as if all of the mass enclosed within a sphere of radius r 0 ...
Hollow sphere of radius r and mass m. = [1] Solid sphere of radius r and mass m. = [1] Sphere (shell) of radius r 2 and mass m, with centered spherical cavity of radius r 1. When the cavity radius r 1 = 0, the object is a solid ball (above).
In geodesy and geophysics, theoretical gravity or normal gravity is an approximation of Earth's gravity, on or near its surface, by means of a mathematical model. The most common theoretical model is a rotating Earth ellipsoid of revolution (i.e., a spheroid ).
A common misconception occurs between centre of mass and centre of gravity.They are defined in similar ways but are not exactly the same quantity. Centre of mass is the mathematical description of placing all the mass in the region considered to one position, centre of gravity is a real physical quantity, the point of a body where the gravitational force acts.
The gravity depends only on the mass inside the sphere of radius r. All the contributions from outside cancel out as a consequence of the inverse-square law of gravitation. Another consequence is that the gravity is the same as if all the mass were concentrated at the center. Thus, the gravitational acceleration at this radius is [14]