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By Chinmony T.
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Sample text
9. 5 Equations of Motion In this paragraph some consequences of Newton’s axioms shall be investigated and applied. 6) the internal forces cancel). As a special case one obtains the corresponding results for a system, which consists of only a single particle. 1 Definitions Definition 9. (for a single particle, characterized by the index i): mi ri pi = mi r˙ i l i = r i × pi Fi Ti = 12 mi r˙ 2i the mass of the particle; the coordinate (the position) of the particle; the momentum of the particle; the angular momentum of the particle with respect to the origin (at r = 0) the force acting on the particle; kinetic (translational22 ) energy23 of the particle.
1) is a differential equation of second order. The general solution contains two integration constants. These integration constants are to be fixed by two initial conditions (boundary conditions). Even though there are no general solution methods, the solution in special cases can be given. 1 Constant Force Let the force be F (x, x, ˙ t) = F0 . Then one can integrate once immediately, t m t dt = m [x(t) ˙ − x(t ˙ 0 )] = F0 (t − t0 ) x ¨ dt = F0 t0 t0 or x(t) ˙ = v0 + F0 (t − t0 ) m ˙ 0 ). A second integration leads to with v0 = x(t 30 2 Newtonian Mechanics: First Applications x(t) = x0 + v0 (t − t0 ) + F0 2 (t − t0 ) 2m with x0 = x(t0 ).
14) and has at the limits of integration α= 0 for ϕ = 0 π/2 for ϕ = ϕ0 . The differential of sin α is d(sin α) = cos α dα ϕ 1 1 − sin 1 cos 2 = ϕ0 dϕ = 2 sin 2 2 sin ϕ20 sin ϕ20 cos α dα . 12) √ √ cos ϕ − cos ϕ0 = 2 sin2 ϕ0 ϕ − sin2 2 2 √ ϕ0 2 sin 1 − sin2 α 2 √ ϕ0 cos α, = 2 sin 2 = 9 An extensive collection of special functions, their properties, tabulated values, and graphical representations can be found in [10]. 40 2 Newtonian Mechanics: First Applications and for the period one obtains ϕ0 8l g T = 0 1 − sin2 0 π/2 l g =4 2 sin ϕ20 cos α dα π/2 8l g = 1 dϕ √ cos ϕ − cos ϕ0 ϕ0 2 1 ·√ 2 sin ϕ20 cos α sin2 α dα 1 − sin2 0 ϕ0 2 .