Q$1).$ A heavy flywheel, rotating initially at $2420$ rpm was allowed to slow down to $1242$ rpm in $9\mathrm{.}2$ min due to air and bearing friction, and was then brought to rest from this speed by the application of a brake which caused an angular deceleration of $\backslash (1\mathrm{.}25\backslash$mathrm$\{$rad$\}/\backslash$mathrm$\{$s$\}^\{2\}\backslash ).$ Calculate the total time in minutes and the total number of revolutions made in coming to rest from the initial speed of $2420$ rpm $.$

Q$2).$ During a test on a large boring mill, its circular worktable, revolving in a horizontal plane, was observed to accelerate from rest at the rate of $\backslash (0\mathrm{.}25\backslash$mathrm$\{$rad$\}/\backslash$mathrm$\{$s$\}^\{2\}\backslash )$ for $26$ s $,$ then rotate at constant speed for $95$ s before coming to rest after a period of retardation lasting $22$ s $.$ Determine the total number of revolutions made by the worktable during the test.

Q$3).$ A spin$-$dryer with a load of washing rotates at $255$ rpm until the power is turned off. It takes $2\mathrm{.}5$ min for it to come to rest. Given that the mass moment of inertia of loaded spin$-$dryer is $\backslash (1\mathrm{.}82\backslash$mathrm$\{$~kg$\}.\backslash$mathrm$\{$m$\}^\{2\}\backslash ),$ calculate the friction torque responsible for bringing the dryer to rest.

Q$4).$ A cast$-$iron pulley, $1\mathrm{.}3$ m in diameter, has a mass moment of inertia of $\backslash (47\mathrm{.}5\backslash$mathrm$\{$~kg$\}\backslash$cdot $\backslash$mathrm$\{$~m$\}^\{2\}\backslash ),$ and average resistance torque of $\backslash (45\backslash$mathrm$\{$~N$\}.\backslash$mathrm$\{$m$\}\backslash ).$ When accelerating from rest, the pulley is subjected to constant belt tensions of $810$ N and $430$ N on tight and slack sides of the belt respectively. Determine how long it takes to accelerate the pulley to $1425$ rpm $.$

Q$5).$ Determine the mass moment of inertia of the flywheel shown in the diagram below if the material is steel. The density of steel is $\backslash (8050\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash ).$