Consider a pendulum with a bob of mass m that is suspended from a spring rather than a rigid rod. Assume that the spring has spring constant k, that it has negligible rest length and mass (that is, you can treat both as zero), that the spring only stretches (it does not bend), and that the pendulum oscillates in a plane (making this a two-dimensional problem). The force due to the spring follows Hooke's law, F =-kr (and there is of course also the gravitational force, with magnitude g, pointing downward). a) (3 points) Write down Newton's second law for this system in Cartesian coordinates in the form of two differential equations for x(t) and y(t). You do not need to solve the differential equations. However, be sure to draw a free-body diagram in which you also define and label your coordinates. b) (4 points) Repeat a), but now in polar coordinates (r, o). Again, you do not need to solve the differential equations for r(t) and o(t), but do draw a free-body diagram and label your coordinates. You also do not need to simplify or decouple the equations; just leave them as one equation related to F, and another related to Fo. c) (4 points) Verify (by substitution) that x(t) = Ax cos(wt + 6r) and y(t) = Ay cos(wt + y) +h satisfy the differential equations that you found in part a) for arbitrary real constants Ar Ay, dr, and dy, and specific constant values of w and h. Determine the values of w and h in terms of the other constants (g, k, and m) of this problem. d) (3 points) Suppose that r = = 0 (which corresponds to no initial velocity) and (0) = y(0)=h/2. What is r(t), the distance from the origin as a function of time? [Hint: Don't attempt to solve the differential equation in part b); try to obtain it from your solution in part c) when specialized to the initial conditions given in this part.]