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A convex lens forms an image of a -tall soda can. The image is tall and appears from the lens.(a) Where is the can? (b) What is the lens's focal length?

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Text: mx=-cx-kx+kx^2-x mx^2=-cx^2-kx^2+kx-x where u=andu^2=2 a. For this part of the problem, we assume no damping, so c=0. Define the natural frequency by the constant w=k. Define new state variables y=[y1 y2 y3 y4], where y1=x1, y2=u1, y3=2, and y4=u2. Rewrite the above system of second-order linear ODEs into a system of first-order linear ODEs: y=Ay, y0=yo (2). Transform A into Ja, where Ja is a matrix in real Jordan canonical form. Show how you obtain your nonsingular transforming matrix, P, and give its inverse, P^-1. (Hint: You may want to use the theorem described on Slide 36 of the Fundamental Solutions lecture notes, which was added after the lecture.) Write the fundamental solution, φ(t) = e^(Ja*t). Dividing R into E (stable), E (unstable), and Eccenter subspaces. With c=0, give the dimension of each of these subspaces for this example. What is the equilibrium point and is it hyperbolic? Briefly explain these implications for the qualitative behavior of this system. Are there limitations to this theory for this particular model? c. Again with c=0, assume this system is initially at rest, and the two masses are displaced by 10 and 20, respectively, i.e., we have: y0=x1(0),0,x2(0),0. Write the unique solution to this initial value problem, y(t). Briefly discuss the solution describing the motion when the two masses are equally displaced to the right, i0=2o>0 (symmetric motion). Also, briefly discuss the solution describing the motion when the two masses are equally displaced in opposite directions, i0=-2o>0 (antisymmetric motion). d. For this part of the problem, we assume damping, c>0. Define ω^2=m^2=w^2, and new state variables y=[y1, y2, y3, y4], where y1=x1, y2=u1, y3=x2, and y4=u2. Rewrite the above system of second-order linear ODEs into a system of first-order linear ODEs: y=By, y0=yo (3). Transform B into Jo, where J is a matrix in real Jordan canonical form. Show how you obtain your nonsingular transforming matrix, P. Write the fundamental solution, φ(t)=e^(Jt).
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Question Text
A convex lens forms an image of a -tall soda can. The image is tall and appears from the lens.(a) Where is the can? (b) What is the lens's focal length?
TopicAll topics
SubjectPhysics
ClassClass 12