2. Consider the torsional bar shown below. T o The rigid bar is massless and of...

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2. Consider the torsional bar shown below. T o The rigid bar is massless and of length L with a vertical force F at its end. A torsional spring at the bar's base has torsion given by T k0, where is the angle of the bar from the vertical, and k is the torsional stiffness of the spring. The angle is sought where the system is in static equilibrium under the action of the force F and torsion T. (a) Apply the principles of static equilibrium to a free-body diagram to show that the governing equation is given by f(0) = k0 FL sin 0 = 0. Observe that this is a nonlinear equation for given k, F, and L. = (b) Evaluate the first two iterations of Newton's method by hand to obtain an estimate of 0 for k= 8, L 10, and F = 1 with an initial guess of 0 (0) 0.9 radians. (You can use a calculator/computer to evaluate your expressions.) (c) Use Python or Matlab, obtain the fully converged root for and compare with your solution for part (b). Here you may either use a built-in function, or write your own (or adapt code from the lectures). (d) Apply two iterations of the bisection method by hand, with an initial in- terval of [4, 5]. Use the same values of k, F, and L as in parts (b)-(c). 2. Consider the torsional bar shown below. T o The rigid bar is massless and of length L with a vertical force F at its end. A torsional spring at the bar's base has torsion given by T k0, where is the angle of the bar from the vertical, and k is the torsional stiffness of the spring. The angle is sought where the system is in static equilibrium under the action of the force F and torsion T. (a) Apply the principles of static equilibrium to a free-body diagram to show that the governing equation is given by f(0) = k0 FL sin 0 = 0. Observe that this is a nonlinear equation for given k, F, and L. = (b) Evaluate the first two iterations of Newton's method by hand to obtain an estimate of 0 for k= 8, L 10, and F = 1 with an initial guess of 0 (0) 0.9 radians. (You can use a calculator/computer to evaluate your expressions.) (c) Use Python or Matlab, obtain the fully converged root for and compare with your solution for part (b). Here you may either use a built-in function, or write your own (or adapt code from the lectures). (d) Apply two iterations of the bisection method by hand, with an initial in- terval of [4, 5]. Use the same values of k, F, and L as in parts (b)-(c).