- Cotes' spiral
In
physics and in themathematics ofplane curve s, Cotes' spiral is aspiral that is typically written in one of three forms:frac{1}{r} = A cosleft( k heta + varepsilon ight)
:frac{1}{r} = A coshleft( k heta + varepsilon ight) :frac{1}{r} = A heta + varepsilon
where "r" and "θ" are the radius and
azimuthal angle in apolar coordinate system , respectively, and "A", "k" and "ε" are arbitraryreal number constants. These spirals are named afterRoger Cotes . The first form corresponds to anepispiral , whereas the third form corresponds to "reciprocal spiral", also known as a "hyperbolic spiral ".The significance of Cotes' spirals for physics are in the field of
classical mechanics . These spirals are the solutions for the motion of a particle moving under a inverse-cubecentral force , e.g.,:F(r) = frac{mu}{r^3}
where "μ" is any
real number constant. A central force is one that depends only on the distance "r" between the moving particle and a point fixed in space, the center. In this case, the constant "k" of the spiral can be determined from μ and theareal velocity of the particle "h" by the formula:k^{2} = 1 - frac{mu}{h^2}
when "μ" < "h" 2 (
cosine form of the spiral) and:k^{2} = frac{mu}{h^2} - 1
when "μ" > "h" 2 (
hyperbolic cosine form of the spiral). When "μ" = "h" 2 exactly, the particle follows the third form of the spiral:frac{1}{r} = A heta + varepsilon.
Bibliography
*
*
Roger Cotes (1722) "Harmonia Mensuarum", pp. 31, 98.*
Isaac Newton (1687) "Philosophiæ Naturalis Principia Mathematica ", Book I, §2, Proposition 9.*
*
Wikimedia Foundation. 2010.