B$.$ Pendulum

A physical pendulum consists of a uniform wooden rod hung from a nail near one end, of length $=0\mathrm{.}89$ml$=0\mathrm{.}89$m$,$ and mass $=0\mathrm{.}42$kgm$=0\mathrm{.}42$kg$.$ The motion is damped because of friction in the pivot, and the damping force is approximately proportional to $$ddt$.$ The rod is set in oscillation by displacing it $0=14$$0=14$ from its equilibrium position and releasing it$.$ After $7\mathrm{.}8$s$7\mathrm{.}8$s$,$ the amplitude of the oscillation has been reduced to $5\mathrm{.}85\mathrm{.}8.$ The angular displacement can be written as $=$cos$$$=$Ae$$tcost$.$

i$.$ Find the value of $$$.[3]$

$=$$=$ s$-1$

ii$.$ Considering that the undamped frequency for this pendulum is $0=$$0=$mghI with $$h the position of the of the centre of mass and $=23$I$=$ml$23,$ find the approximate period of the motion.$[2]$

$=$T$=$ s

iii. Determine how long it takes for the amplitude to be reduced to $1212$ of its original value.$[2]$

$=$t$=$ s