Page - 125 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

4.3 OscillatingOne-DimensionalSystems 125 Fig.4.15 Sketchof aone-dimensional, oscillatingdynamic system(without friction) of thebodyon thex axis, alongwhich thebodymoves. Thespring isnot stretched whenx D 0, so the force is zero, andx D 0 is hence the equilibriumposition of the body. The spring force is kx, wherek is a constant to bemeasured. We as- sume that there arenoother forces (e.g., no friction). Newton’s 2nd lawofmotion F Dma thenhasF D kx andaD Rx, kxDmRx; (4.41) whichcanbe rewrittenas RxC!2xD0; (4.42) by introducing!Dpk=m (which isverycommon). Equation (4.42) is a second-order differential equation, and thereforewe need two initial conditions, oneon thepositionx.0/andoneon thevelocityx0.0/. Here wechoose thebody tobeat rest, butmovedaway fromits equilibriumposition: x.0/DX0; x0.0/D0: Theexact solutionof (4.42)with these initial conditions isx.t/DX0cos!t. This caneasilybeverifiedbysubstituting into (4.42)andchecking the initial conditions. The solution tells that such a spring-mass system oscillates back and forth as de- scribedbyacosinecurve. Thedifferential equation (4.42)appears innumerousother contexts. Aclassical example is a simplependulumthatoscillates backand forth. Physicsbooksderive, fromNewton’s second lawofmotion, that mL 00Cmgsin D0; wherem is themass of the bodyat the endof a pendulumwith lengthL,g is the acceleration of gravity, and is the angle the pendulummakeswith the vertical. Consideringsmall angles , sin , andweget (4.42)withxD ,!Dpg=L, x.0/ D , andx0.0/ D 0, if is the initial angle and the pendulum is at rest at t D0.