QUESTION $1$

Consider the system shown below that has a Cartesian $\backslash ($ O x y $\backslash )$ coordinate system fixed at $\backslash ($ O $\backslash )$ where the $\backslash ($ x $\backslash )-$axis is horizontal and the $\backslash ($ y $\backslash )-$axis is vertical. The rotational displacement $\backslash (\backslash$theta$($t$)\backslash )$ of the bar is positive in an anti$-$clockwise direction.

For the above system determine the following information clearly showing all steps and appropriate reasoning:

$1\mathrm{.}1$ Determine the differential equation of motion of the bar as it oscillates about its pivot if the mass of the bar is $7\mathrm{.}3$ kg ;

$1\mathrm{.}2$ Solve the differential equation of motion of the bar as it oscillates about its pivot;

$1\mathrm{.}3$ Plot a graph of the solution of the differential equation inclusive of the rotational displacement and the rotational velocity of the motion of the bar as it oscillates about its pivot for a time corresponding to the natural angular frequency;

Total Marks:

$[50]$

QUESTION $2$

Consider the system shown below that has a Cartesian $\backslash ($ O x y $\backslash )$ coordinate system fixed at $\backslash ($ O $\backslash )$ where the $\backslash ($ x $\backslash )-$axis is horizontal and the $\backslash ($ y $\backslash )-$axis is vertical. The rotational displacement $\backslash (\backslash$theta$($t$)\backslash )$ of the bar is positive in an anti$-$clockwise direction.

For the above system determine the following information clearly showing all steps and appropriate reasoning:

$2\mathrm{.}1$ Determine the differential equation of motion of the bar as it oscillates about its pivot if the mass of the bar is $7\mathrm{.}3$ kg ;

$2\mathrm{.}2$ Solve the differential equation of motion of the bar as it oscillates about its pivot;

$2\mathrm{.}3$ Plot a graph of the solution of the differential equation inclusive of the rotational displacement and the rotational velocity of the motion of the bar as it oscillates about its pivot for a time corresponding to the natural angular frequency;