B$2.($a$)$ At a particular instant an airplane is diving downwards at a $30$ degrees angle with respect to the horizontal. It's engine is off and it is travelling at a speed of $\backslash (800\backslash$mathrm$\{$~km$\}/\backslash$mathrm$\{$hr$\}\backslash ).$ Due to aerodynamic lift, its instantaneous acceleration is $3\mathrm{.}0$ g in the upward direction, where $\backslash ($ g $\backslash )$ is gravitational acceleration on Earth surface.

$($i$)$ Determine the rate of speed change of the airplane.

$(2$ marks$)$

$($ii$)$ Determine the radius of curvature of the path.

$(3$ marks$)$

$($b$)$ By starting from first principles and performing an integration, determine the mass moment of inertia of a uniform bar of mass $\backslash ($ m $\backslash )$ and length $\backslash ($ L $\backslash )$ about an axis through the center perpendicular to the length. Describe how the moment of inertia changes if the rotation axis is moved to the end of the bar.

$(5$ marks$)$

$($c$)$ The figure below shows a tilting device that maintains a sloshing water bath for washing vegetable produce. The crank OA oscillates about the vertical axis.

$($i$)$ The crank OA has a clockwise angular velocity of $\backslash (4\backslash$pi $\backslash$mathrm$\{$rad$\}/\backslash$mathrm$\{$s$\}\backslash )$ when it is exactly vertical. Determine the angular velocity of the basket in the position shown where $\backslash (\backslash$theta$=30\backslash )$ degrees.

$($ii$)$ If the angle $\backslash (\backslash$theta $\backslash )$ is increased to $45$ degrees, describe how the angular velocity of the basket in this position is affected.