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- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Tatum)/18%3A_The_CatenaryIf a flexible chain or rope is loosely hung between two fixed points, it hangs in a curve that looks a little like a parabola, but in fact is not quite a parabola; it is a curve called a catenary, whi...If a flexible chain or rope is loosely hung between two fixed points, it hangs in a curve that looks a little like a parabola, but in fact is not quite a parabola; it is a curve called a catenary, which is a word derived from the Latin catena, a chain.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/05%3A_Calculus_of_Variations/5.09%3A_Lagrange_multipliers_for_Holonomic_ConstraintsThe Lagrange multiplier technique provides a powerful, and elegant, way to handle holonomic constraints using Euler’s equations. The general method of Lagrange multipliers for n variables, with m ...The Lagrange multiplier technique provides a powerful, and elegant, way to handle holonomic constraints using Euler’s equations. The general method of Lagrange multipliers for n variables, with m constraints, is best introduced using Bernoulli’s ingenious exploitation of virtual infinitessimal displacements, which Lagrange signified by the symbol δ.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Tatum)/18%3A_The_Catenary/18.03%3A_Equation_of_the_Catenary_in_Rectangular_Coordinates%2C_and_Other_Simple_RelationsIf one end of the chain is fixed, and the other is looped over a smooth peg, Equation ??? shows that the loosely hanging vertical portion of the chain just reaches the directrix of the c...If one end of the chain is fixed, and the other is looped over a smooth peg, Equation ??? shows that the loosely hanging vertical portion of the chain just reaches the directrix of the catenary, and the tension at the peg is equal to the weight of the vertical portion.
- https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/19%3A_Statics/19.03%3A_The_Catenarywhere a=H/w,H is the horizontal tension in the chain at the pole (in newtons), and w is the linear weight density of the chain (in newtons per meter). Note that if the horizontal tension \(...where a=H/w,H is the horizontal tension in the chain at the pole (in newtons), and w is the linear weight density of the chain (in newtons per meter). Note that if the horizontal tension H is very large (the chain is pulled very taut), then a=H/w is very large, d/2a is very small, and so sinh(d/2a)≈d/2a, so that st≈d, as expected.
