A merry$-$go$-$round consists of a solid disc $($platform$),$ a thin railing, and $14$ horses, arranged in a two rows. The platform has mass $M_p=400kg$ and radius $R_p=3m.$ The railing with total mass $M_r=100kg$ runs along the edge of the platform, so $R_r=3m.$ Each of the horses has mass $M_h=25kg.$ Six of the horses are at inner radius $R_i=1m,$ and the other eight are at outer radius $R_0=2m.$

The merry$-$go$-$round rotates rigidly about the axis through its centre. At $t=0$ the merry$-$go$-$round is initially at rest and it begins to accelerate at $\frac{1}{3}ra\frac{d}{s^2}.$ This acceleration lasts until $t=6$ s at which it drops to zero, and the merry$-$go$-$round continues to rotate at a constant angular velocity.

$($a$)($i$)$ Express the angular velocity of the merry$-$go$-$round during the time interval $0t6s.$

$($ii$)$ Determine the angular velocity of the merry$-$go$-$round for times after $t=6s.$

$($iii$)$ Express the angular displacement $[$from its rest position$]$ during $0t6s.$

$($iv$)$ Determine the angular displacement for times after $t=6s.$

$($c$)($i$)$ Modelling the platform as a uniform circular disc, compute its moment of inertia.

$($ii$)$ Modelling the railing as a thin ring, compute its moment of inertia.

$($iii$)$ Modelling each inner horse as a point mass, compute the moment of inertia of one inner horse.

$($iv$)$ Modelling each outer horse as a point mass, compute the moment of inertia of one outer horse.

$($v$)$ Compute the total moment of inertia of the merry$-$go$-$round.

$($d$)$ Determine the value of the net torque $[$about the axis of symmetry$]$ acting on the merry$-$go$-$round while it accelerated.