Question

Pls, use Mathlab to answer all questions.

Thank you.

2. The energy of the mass-spring system is given by the sum of the kinetic energy and the potential energy. In the absence of damping, the energy is conserved mu2ky2 (a) Add commands to LAB05ex1 to compute and plot the quantity E as a function of time. What do you observe? (pay close attention to the y-axis scale and, if necessary, use ylim to get a better graph). Include at least one plot. Does the graph confirm the fact that the energy is conserved? (b) Show analytically that dE 0. (Note that this proves that the energy is constant) (c) Add commands to LAB05 ex1 to plot curve ever get close to the origin? Why or why not? What does that mean for the mass-spring system? v vs y (phase plot). Include the plot. Does the

Answer 1

Please find attached the copy of the matlab script LAB05ex1a.m below with parts (a) and (c) completed in the script along with the plots.

On line 9 added Command E mv2 +ky2 On line 10 appended the plot command with the Following parameters t, E, g The observaticn is that the energy curve hori zontal line i.e. Energy (ie. total- 0 S! energy) is constant w.r.t time and d not change with time. Hence the graph confirms the pact that energy is conserved. since, E (b) 2 Differentiate Ew.Y.t tgives dE dy +k 2y m.2v dv dt dt From equation opF motion

mdv-ky dE dt 2 (mv) dv dt 2(ky) v 2 dy dt ( 5mdv --ky) dt Hen ce proved (C) On line 15 added the command to plot figure (2) plot (y, v) he curve hever gets close to the origin because energy is conserved so the motion never dampens ie. amplitude ef the motion does not decyeases with time. Hence this Spring-mass energy conserved, System has its

LABO5exla.m x clear close all; clc; m 1 Value Name 1 - 109x1 double 2- k=4: 3 - m omegao C= 1: 4- omega0 sqrt(k/m); p = c/(2*m) y0 = -0.8; v0 = -0.3; t 0.5000 5 - 109x1 double 109x1 double 6 - ode45(af, [o, 101, [y0, v0], [, omega0, p); 109x1 double 109x2 double E 1/2 m v.^2 + 1/2 k* y.^2; % energy [J figure (1) plot( t, y, 'bo-', t, v, 'r+-', t, E, 'g'); xlabelt legend'y'v', 'E') grid on; axis tight; 9 yo -0.8000 10 11. 12 - 13 . 14 figure (2) plot( y, v ) xlabel'y' ylabel'v') grid on axis square; 15 16. 17 18 f(t, Y, omega0, p) function dYdt 19 20 y Y(1); v = Y(2); a = -omega0^2 * y; dYdt [v ; a] 21 - end 22 Command Window New to MATLAB? See resources for Getting Started, fx> Ln 6 Col 22

0.5 0 1.5

2 1.5 1 0.5 C -0.5 1 -1.5 1 0.8 0.6 -2 -1 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 y A