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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} .
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.
For such two- or restricted three-body problems as its simplest examples—e.g., one more massive primary astrophysical body, mass of m1, and a less massive secondary body, mass of m2—the concept of a Hill radius or sphere is of the approximate limit to the secondary mass's "gravitational dominance", [6] a limit defined by "the extent" of its ...
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]
rad/s is the diurnal angular speed of the Earth axis, and km the radius of the reference sphere, and the distance of the point on the Earth crust to the Earth axis. [ 3 ] For the mass attraction effect by itself, the gravitational acceleration at the equator is about 0.18% less than that at the poles due to being located farther from the ...
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 was concentrated at the center of the mass distribution (as noted above). The portion of the mass that is located at radii r > r 0 exerts no net gravitational force at the radius r 0 from
All the mass at a greater distance than r from the center has no resultant effect. For example, a hollow sphere does not produce any net gravity inside. The gravitational field inside is the same as if the hollow sphere were not there (i.e. the resultant field is that of all masses not including the sphere, which can be inside and outside the ...
The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteristic radius associated with any quantity of mass.