STRUCTURES IN EQUILIBRIUM (5 points)

I posted a handout on the theory of structures in equilibrium – pretty much the same as what I described in class that one day. For extra credit I would like you to

(1) Verify that the null vectors I gave for the 3-beam roof are always null vectors, whatever the locations of the beam ends. How would you generalize them for an arbitrary structure?

(2) For the 6-beam roof at the bottom of the handout, let's make it specific by assuming that the 4th corner is at (1,-1)^T. Also imagine you make it a 4-beam quadrilateral by removing the diagonal beams "c" and "e". Find the matrix for this configuration, and find a fourth null vector corresponding to a flexible deformation of the structure.

(3) Reinsert beam "c" and solve for the (hopefully unique) equilibrium, assuming the weight on the top is 600 pounds and a weight of 200 pounds on the bottom vertex (perhaps a hanging chair with a big guy in it).

(4) Reinsert beam "e" and solve for the equilibrium. I suspect you will find that tau_e will be a free variable, and I'd like you to express the other tensions in terms of tau_e. If tau_e=0, do you recover the solution in part (3)?

FUNCTION SPACES (5 points)

A function f(x) defined on an interval of values can be thought of as an infinite-dimensional vector , with the “entries” being the values taken by f(x). We can, if we wish, define infinite dimensional matrices as functions of two variables A(x,y), with multiplication defined by integration instead of summation: . (Though in fact the subject of functional analysis deals with “matrices” or linear operators that can’t always be represented by functions of two variables.) The concepts of dot product and length can be easily generalized: , , and the functions f and g are orthogonal if .

(1) Function approximation: suppose we want to approximate the function f(x)=sin(x) by a cubic polynomial g(x)=, for x in the range from 0 to . We know that the Taylor polynomialdoes a very good job for x close to zero, but gets less accurate for larger values of x. Find the values of a and b for which g(x) is the best approximation to sin(x), and see how close they are to the Taylor values 1 and -1/6 .

(2) Differential equations: suppose we want to solve the initial-value problem withand, for x in the range from 0 to 2.

(a) Find the exact solution in terms of exponentials.

(b) If we did not know about exponentials, we could seek an approximate solution in the form , which satisfies the initial conditions for any value of k. Find the value of k that minimizes (using 0 and 2 as limits of integration).

(c) Graph the exact and approximate solutions and see how close they are.

GEOMETRY OF DETERMINANTS (5 points)

I drew on the board the parallelogram whose edges are the column vectors of the matrix , whose area is the determinant . I also drew a rectangle whose sides are of length a and d, and cut out of it a smaller rectangle of sides b and c (assuming the dimensions work out that way), resulting in a funny shape with the same area.

Is there a simple geometric way to see that these shapes always have the same area? One way would be if you could cut up one of the shapes into pieces, and then reassemble the pieces into the other shape. But it’s rather tricky to figure out a way that works for a variety of values of a, b, c, and d. (I have one that I think works, but I want to see what YOU come up with!)