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We start by setting a = 13 cm, c = 17 cm, and rad/s. We will change these values later. 1. Plot the graph of b against θ, over the range 0 ≤ θ ≤ 2π. 2. Plot the graph of against θ, over the range 0 ≤ θ ≤ 2π. d2b 2. Plot the graph of against θ, over the range 0 ≤ θ ≤ 2π. dt2 3. Using your answers to the previous items, determine the angle θ when the shaft is moving at its fastest. What is that largest speed? 4. How many seconds does it take for the contraption to return to its original state? Justify. 5. For this final item we change the values of a and c, subject to the following requirements: a + c = 30; and 2 ≤ c − a ≤ 10: (a) What is the smallest possible value for a? What is the largest possible value for a? Justify. (b) Create a table of values for a, starting with the smallest possible value, and ending with the largest possible value, in increments of 0.1. For each value a in this table, find the maximum value that can take, and the corresponding angle θmax. Store these values. (c) Plot the graph of the maximum values for you found in the last item, against the values of a from the table. Use your plot to determine which value of a will produce the largest maximum speed for the piston. (d) Plot the graph of θmax against a. Which angle θmax corresponds to the (maximum) value of a you obtained in the last item?

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clear all;
a = 0.13;
c = 0.17;
k = (2*pi)/3;
%Ques 1
theta = 0:.01*pi:2*pi;
b = a*cos(theta)+sqrt(c^2-(a*sin(theta)).^2);
figure;plot(theta,100*b,'--k');xlabel('\theta');ylabel('b (cm)');
title('Plot of b vs \theta');

%Ques 2
dbdtheta = -a*sin(theta).*(1+a*cos(theta)./sqrt(c^2-(a*sin(theta)).^2));

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