2. A simple pendulum has a particle of mass m at the end of a light rod of length [. The other end of the rod is attached to a fixed point O, at the origin of polar coordinates (r,#). The particle is at position (r,#) with # = 0 corresponding to the particle being vertically below 0. (a) Use the formulae for acceleration in polar coordinates, a = (i r6*)f + (270 + r0)8 (2) to show that . . " T = mgcosf + ml6?, and 0 = % sin @, where T is the tension in the rod. Use the relation %{92) 266 to deduce that 9 = 27'9 cosf+ A, where A is a constant. If the particle is instantaneously at rest ( = 0) when the rod is horizontal, find # and T when the rod is vertical. How does the tension in the vertical position (which is also the maximal tension) depend on the rod's length [? (b) Assume the particle is subject to linear air resistance av. Use the expression of the velocity in polar coordinates v=rt+rbf (3) together with (2) to write the equations of motion in polar coordinates. Show that, under the small-angle approximation sinf# = #, the particle's position # satisfies the fundamental equation of damped harmonic motion