Consider the mechanical system shown in Figure $1.$ Two identical cylindrical rollers of mass

$m$ and radius $r$ are fixed at their centres. They are free to turn, without friction, in their

bearings, as a uniform rigid beam also having mass $m,$ moves over them without slipping.

Attached to the midpoint of the beam at A is a simple pendulum $($i$.$e$.,$ one with all its mass

concentrated at the tip$)$ of length a and mass $6m.$ Inertia properties for a circular disc are

provided on the following page.

Figure $1$: Schematic illustration of pendulum and beam system.

Perform the following:

$($a$)$ Identify the degrees of freedom in the system and assign appropriate generalized coor$-$

dinates.

$($b$)$ Derive expressions for the position, velocity, and acceleration of the centre of mass of

the beam and the centre of mass of the pendulum.

$($c$)$ Derive expressions for the kinetic energy and potential energy of the system.

$($d$)$ Derive the nonlinear equations governing the motion of the system using either Newton$-$

Euler equations or Lagrange's equation and express the resulting equations in the

appropriate standard form.

$($e$)$ Briefly outline the process that would be used for obtaining a numerical transient

solution to the nonlinear equations of motion based on Euler integration $($assuming

parameter values and initial conditions were provided$).$