w04 d01 lab computational methods

7/27/2019 W04 D01 LAB Computational Methods

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W04_D01

Computational Methods in Physics

PHYS168


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Plot in FreeMat

Graph 2-D data with linear scales for both axes

Basic Command Syntax:

plot(,{linespec 1},,{linespec 2}...,properties...)

**In general, the linespec is composed of three optional parts, the colorspec,the symbolspecand the linestylespecin any order. Each of thesespecifications is a single character that determines the correspondingcharacteristic.

Example:

X = linspace(-pi, pi);

Y = cos(X);

Plot (X, Y, r*-)


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Plot in FreeMat

Example:

X = linspace(-pi, pi);

Y = cos(X);

Plot (X, Y, r*-)


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Plot in FreeMat

colorspec:

'b' - Color Blue

'g' - Color Green

'r' - Color Red

'c' - Color Cyan

'm' - Color Magenta

'y' - Color Yellow

'k' - Color Black


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Plot in FreeMat

The symbolspecspecifies the (optional) symbol to bedrawn at each data point:'.' - Dot symbol

'o' - Circle symbol

'x' - Times symbol

'+' - Plus symbol

'*' - Asterisk symbol

's' - Square symbol

'd' - Diamond symbol

'v' - Downward-pointing triangle symbol

'^' - Upward-pointing triangle symbol

'' - Right-pointing triangle symbol


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Plot in FreeMat

The linestylespecspecifies the (optional) line style

to use for each data series:

'-' - Solid line style

':' - Dotted line style

'-.' - Dot-Dash-Dot-Dash line style

'--' - Dashed line style


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Plot in FreeMat

Example: SUPERIMPOSE MULTIPLE GRAPHS IN ONE CANVAS

x = linspace(-pi,pi); y = [cos(x(:)), cos(3*x(:)), cos(5*x(:))]; plot(x,y)


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Plot in FreeMat

Example: SUPERIMPOSE MULTIPLE GRAPHS IN ONE CANVASx = linspace(-pi,pi); y = [cos(x(:)), cos(3*x(:)), cos(5*x(:))];

plot(x, y(:,1), 'rx-', x, y(:,2), 'b>-', x, y(:,3), 'g*:');


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Plot in FreeMat

Example: for complex matrices y = cos(2*x) + i * cos(3*x);

plot(y);

plot(real(y))

plot(imag(y))


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Plot in FreeMat

Example: for complex matrices t = linspace(-3,3);

plot(cos(5*t).*exp(-t),'r-','linewidth',3);


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Plot in FreeMat

Other tweaks in plotting graphs

Title

title( title here )

Axes labels axes Range

xlabel( put x-axis label here ) axis([ 0, 10, -1, 1])

ylabel( put y-axis label here )

Legends

legend(legend 1,legend 2,)


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Plot in FreeMat

Multiple graphs in a canvas (NOT Superimposed)

x = linspace(-pi, pi);

subplot(2,2, 1);plot( x, cos( x ) );

subplot(2,2, 2)

plot( x, cos(2*x(:) ) );

subplot(2,2, 3);

plot( x, cos(5*x(:) ) );

subplot(2,2, 4);

plot( x, cos(10*x(:) ) );


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Saving Variables in a File in FreeMat

Saving matrices or variables into an external file

Syntax:

save(filename,variable1,variable2,, options)

Example: THIS IS IN BINARY FORMAT

savefile = 'pqfile.mat';

p = rand(1, 10);

q = ones(10);

save(savefile, 'p', 'q')


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Loading Variable from a File in FreeMat

Loading matrices or variables from an external file

Syntax:

load(filename)

load filename

Example: savefile = 'pqfile.mat';

p = rand(1, 10);

q = ones(10);

save(savefile, 'p', 'q')

To load the variables p and q:load pqfile.mat

load pqfile.txt

After that we can then access the variables directly;

example disp(p) or disp(q)


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Problem: Underdamped LRC circuit % Consider the LRC circuit as shown we want to analyze this LRC circuit's

% behavior using Kirchhoff's Rules. First we close the switch S in the

% upward position connecting the capacitor to an emf (E) source for a long time

% to ensure that the capacitor acquires its final charge Q=EC. Then at t=0,

% we turn the switch on, applying Kirchhoff's Loop Rule, with clockwise as the

% direction of travel, we can obtain :

% sum( V ) = -iR -L (di/dt) - q/C = 0 [1]

% where i is the current flowing in the circuit at time t, q is the accumulated

% charge in the capacitor at a certain time, R i the resistance of the resistor

% L is the inductance of the inductor and C is the capacitance of the capacitor.

% Substituting i = dq/dt, abouve eqn [1] becomes

% d^2q/dt^2 + (R/L)(dq/dt) + q/(LC) = 0 [2]

% There are three possible solutions to eqn [2]. The first solution corresponds

% to the situation where (R/2L)^2 < (1/LC), also known as the underdamped case.

% The two remaining cases are the overdamped case, where (R/2L)^2 > (1/LC),

% and the critical damped case, where (R/2L)^2 = (1/LC).

% OUR interest here is on the underdamped case, where the solution to eqn [2]

% is actually a clear combination of exponential decay and sinusoidal function,

% as can be seen in the eqn [3] below

% q(t) = ( A exp[(-R/2L)t] ) cos( [sqrt( 1/LC - (R/2L)^2 ) * t + phi ] ) [3]

% note charge is a function of time, let time = 0 10

TASK: plot the sinusoidal exponential decay and exponential decay-only cases.


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Problem 1

Consider the resonant behavior of a forced

oscillation. As discussed in many introductory

physics textbooks, resonance occurs whenever

the frequency of the driving force matcheswith the natural frequency of the physical

system. If the driving force is periodic, the

oscillation is expressed as dierential equation,


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Problem 1, contd where m is the mass, k is the spring constant, b is the

damping parameter, Fe is the amplitude of the drivingforce and we is the frequency of the periodic drivingforce. The solution to this

dierential equation yields,

where u = we

/w and w is the natural frequency of thephysical system.Obtain plots of the amplitude (Fe/G) of Equation 1.6versus we=w for b = 0.100 , b = 0.200 and b =0.400 . Use the following values: F

e

= 10, m = 1 and k= 1

w04 d01 lab computational methods

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