(Solved): Planar Pendulum Consider a planar pendulum which consists of a mass, m, suspended by a rigid ( mas ...

Planar Pendulum Consider a planar pendulum which consists of a mass, $\mathrm{m}$, suspended by a rigid ( massless) rod of length, l. Gravity acts in the downward vertical direction. The position of the mass can be described by the angle, $?$, or by the Cartesian coordinates $(\mathrm{x},\mathrm{y})$. a) Formulate equations of motion using Cartesian coordinates and an appropriate geometric constraint equation (as done with the rollercoaster). The solutions to the equations of motion and the constraint equation will yield the positions ( $\mathrm{x}$ and $\mathrm{y}$ ) and velocities of the mass, and the tension in the rod. Write these equations in first-order, state-variable form that can be given to ODE45. There will be five equations. b) Formulate equations of motion using a body-fixed reference frame (e.g., cylindrical coordinates). Solution to the equations of motion will yield the angular position and velocity of the mass and the tension in the rod. Why is a separate constraint equation not needed here? Show that these equations are equivalent to those in part a. c) Using angular momentum about the pivot, derive a second-order differential equation that governs $?$ (will be the same as the one found in part $b$ ). d) Create a MATLAB simulation of the pendulum. Overlay plots of both the trajectory and the angular position of the mass calculated by both sets of equations (parts $a$ and $b$ ) for a chosen (by you) set of parameters and initial conditions (specify a position appropriate to the rod length, 1 , and zero velocity). e) Using your simulation calculate (using states from either coordinate system) and plot the tension force in the rod, and the kinetic, potential, and total energy of the system as functions of time for your selected case.