Problem 1 Fou r point objects of equal mass m are placed at coordinates: {{9: B, a} s {99 a: a} s {3: B, a}: {3: 3: 3}}; a) Calculate the inertia tensor of this assembly of four point-masses. (hint: it's simple to map the T function (dened in lectures) across the list of points, and sum the resulting matrices}. b) Calculate the principal moments (eigenvalues) and principal axes (eige nvectors) by solving the secular determinant. Do this by hand to demonstrate that you know how, but feel free to also verify your result with a computer. c) you should nd that the eigenvectors you calculated are not all automatically orthogonal. Verify that, and x it by constructing from them an orthonormal set of eigenvectors. You can use Gram- Schmidt orthogonalization or another method of your choice. Verify that this new set of vectors is mutually orthogonal, normalized, and check that each eigenvector in the set satises the appropriate eigenvalue equation. d) construct a new matrix R whose rows are made of the three eigenvectors. R can be viewed as a rotation matrix that rotates the original coordinate system to a new one that is aligned with the princi- pal axes. Show that the matrix product R.I.Tra nsposelR] is diagonal, with diagonal elements equal to the principal values. That's the inertia tensor expressed in this new \"aligned\" coordinate system. e) use the expression derived in lecture (it's not in book) in terms of the T matrix and center of mass to shift the origin from the corner {0,0,0} to the center of mass of the objects. Remember the total mass M = 4 m. Show that the inertia tensor in this shifted coordinate system is proportional to the identity matrix, and find the values of the (equal) diagonal elements