Solver Initial Guess Root Ea Iterations Falsepos 50 39.0566 2.16e-06 10 Bisect 50 39.0566 3.94e-07 6 Mod_Secant 50 39.0568 9.92e-06 624 Newtraph 50 39.0566 7.63e-06 24

Divergence of Newton-Raphson Method Iteration x_i Approximate Error 0 2.0000 NaN 1 2.2857 1.25e+01 2 2.5276 9.57e+00 3 2.7422 7.83e+00 4 2.9375 6.65e+00 5 3.1182 5.79e+00

Convergence of Newton-Raphson Method Iteration x_i Approximate Error 0 0.200000 NaN 1 -0.017391 1.25e+03 2 0.000011 1.65e+05 3 -0.000000 4.51e+11 4 0.000000 4.51e+11 5 0.000000 4.51e+11

#HW_4

Collar Potential

function [PE] = collar_potential(x,theta)

X=(.5-sqrt(.5^2 + (.5-x)^2)); PE_s=15X^2; PE_g=4.905x*sind(theta);

PE=PE_g + PE_s;

end

HW_4 Script

%Potential Energy Minimization Script

a=1; for theta=1:90; [x,fx,ea,iter]=goldmin(@(x) collar_potential(x,theta),.01,1,.0001,100); X(a)=x; Theta(a)=theta; FX(a)=fx; a=a+1; end X(a)=X(a-1); Theta(a)=Theta(a-1); plot(Theta,X) title('Potential Energy Optimization') xlabel('Theta (Degrees)') ylabel('x_c (meters)')