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A parametric equation for the sphere with ... Archimedes first derived this formula by showing that the volume inside a sphere is twice ... (the ratio of mass to ...
In all cases, the equations are collectively called a parametric representation, [2] or parametric system, [3] or parameterization (also spelled parametrization, parametrisation) of the object. [ 1 ] [ 4 ] [ 5 ]
Dividing both force equations by the respective masses, subtracting the second equation from the first, and rearranging gives the equation ¨ = ¨ ¨ = = (+) where we have again used Newton's third law F 12 = −F 21 and where r is the displacement vector from mass 2 to mass 1, as defined above.
For example, one sphere that is described in Cartesian coordinates with the equation x 2 + y 2 + z 2 = c 2 can be described in spherical coordinates by the simple equation r = c. (In this system—shown here in the mathematics convention—the sphere is adapted as a unit sphere, where the radius is set to unity and then can generally be ignored ...
The great-circle distance, orthodromic distance, or spherical distance is the distance between two points on a sphere, measured along the great-circle arc between them. This arc is the shortest path between the two points on the surface of the sphere. (By comparison, the shortest path passing through the sphere's interior is the chord between ...
It consists of a mass m moving without friction on the surface of a sphere. The only forces acting on the mass are the reaction from the sphere and gravity . Owing to the spherical geometry of the problem, spherical coordinates are used to describe the position of the mass in terms of ( r , θ , ϕ ) {\displaystyle (r,\theta ,\phi )} , where r ...
The second equation says that the kinetic energy (seen from the origin) of a particle of unit mass moving with the expansion plus its (negative) gravitational potential energy (relative to the mass contained in the sphere of matter closer to the origin) is equal to a constant related to the curvature of the universe. In other words, the energy ...
The added mass can be incorporated into most physics equations by considering an effective mass as the sum of the mass and added mass. This sum is commonly known as the "virtual mass". A simple formulation of the added mass for a spherical body permits Newton's classical second law to be written in the form