* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Instructions-damped-SHM
Monte Carlo methods for electron transport wikipedia , lookup
Mean field particle methods wikipedia , lookup
Jerk (physics) wikipedia , lookup
Analytical mechanics wikipedia , lookup
Renormalization group wikipedia , lookup
Wave packet wikipedia , lookup
Lagrangian mechanics wikipedia , lookup
Path integral formulation wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Classical mechanics wikipedia , lookup
Seismometer wikipedia , lookup
Spinodal decomposition wikipedia , lookup
Matter wave wikipedia , lookup
Routhian mechanics wikipedia , lookup
Work (physics) wikipedia , lookup
Newton's theorem of revolving orbits wikipedia , lookup
Hunting oscillation wikipedia , lookup
Newton's laws of motion wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Brownian motion wikipedia , lookup
Centripetal force wikipedia , lookup
Damped Simple harmonic motion A particle of mass m is undergoes oscillatory motion if, on displacement from its equilibrium position, it experiences a restoring force proportional to its displacement – the well known simple harmonic motion. The equation of motion is d 2x m 2 m 2 x, dt where is the natural frequency of oscillation. If there is also a drag force -mv that is proportional to the instantaneous speed v the equation of motion becomes m where is a drag coefficient. d 2x dx m 2 x m , 2 dt dt The earlier exercise on derivatives showed how to get a numerical approximation to the first derivative. For x tabulated at intervals of t we have dx xn 1 xn dx xn 1 xn 1 or . t 2 t dt dt The second form was found to converge more quickly so we will use it here. The second derivative can be approximated by dx dx 2 d x dt n dt n 1 dt 2 t xn 1 xn xn xn 1 2 d x t t xn 1 2 xn xn 1 . 2 dt 2 t t Thus the equation of motion may be approximated by xn1 2 xn xn1 x x 2 xn n1 n1 . 2 2 t t Thus we obtain 2 2 t 2 xn xn1 1 t / 2 xn1 . 1 t / 2 This equation can be used to generate the positions xn 1 from xn and xn 1 . Thus we can obtain positions x2 and onwards. However x1 cannot be obtained this way as it requires a position x1 before the start! The effective initial acceleration, at t = 0, in the x-direction is 2 x 0 v 0 . We now have a two distinct choices; either the particle is situated at the origin and given an initial velocity along the x-axis ( x 0 0; v 0 v0 ), or it is displaced from the origin by an amount a (the amplitude) and then released ( x 0 a; v 0 0 ). We’ll choose the latter case here. Thus using the equation “ s ut Construct a spreadsheet which contains labelled cells for the quantities a, , and t. You may find it convenient to calculate the subsidiary quantities t 1 2 ft ” we can approximate 2 2 1 x1 a 2 a t . 2 2 , 2 2 t , 1 t / 2 , 1 t / 2 to avoid repeated calculation of 2 these quantities for each position. If these are positioned near the top-left of the spreadsheet, then below them under headings n, t and x compute the position of the particle for integer values of n from 0 to 100. Take t = 0.1 s and initially set = 0 s-1, i.e. no drag. Plot x against t. You should see the familiar cosine curve. Now set 0.4 . The plot should look similar to that below with only the oscillatory curve shown. x-t plot 2.5 2 1.5 1 x 0.5 0 -0.5 0 2 4 6 8 10 -1 -1.5 -2 -2.5 t Add the amplitude curves with the equations a exp t / 2 . Save your spreadsheet as “username-dampedSHM”. Vary the parameter through the range 0 to 4 and note the behaviour of the plotted quantities.