Question $6$

$($a$)$ The uniform disk, shown in Figure $6,$ has a radius $\backslash ($ r$=120\backslash$mathrm$\{$~mm$\}\backslash ).$ The disk is welded at its centre to two elastic rods of equal length with fixed ends at $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash ).$ The disk twists through an $\backslash (8^\{\backslash$circ$\}\backslash )$ angle when a $0\mathrm{.}50$ Nm couple is applied to it in the horizontal plane and in an anti$-$clockwise direction. When the couple is removed, the disk oscillates with a period of $1\mathrm{.}3$ s $.$

$($i$)$ Find the torsional stiffness $($torsional spring constant$)$ of the elastic rods.

$($ii$)$ Draw the free$-$body diagram and the kinetic diagram of the disk after the couple is applied. Hence, use equation of motion to derive an equation is in the 'standard form' for the vibration of the disk.

$($iii$)$ Calculate the natural frequency $\backslash (\backslash$left$(\backslash$omega$\_\{$n$\}\backslash$right$)\backslash )$ of the vibration.

$($iv$)$ Hence or otherwise, determine the mass of the disk.

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$($b$)$ A $\backslash (2-\backslash$mathrm$\{$kg$\}\backslash )$ mass is suspended from a spring, having a spring stiffness $($spring constant$)$ of $\backslash (200\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$m$\}\backslash ).$ The mass is displaced slightly and released. Determine the frequency $($number of cycles per second$)$ of the resulting vibration.