A small ball bearing oscillates with simple harmonic motion along the shaped groove. The ball bearing starts from the right end of the groove. Treat going right as positive.

$($i$)$ If the period is T$,$ write an expression for the displacement x in terms of T$,$ d and t the time, given that the ball bearing started from the top, as shown in the diagram.

$($ii$)$ Calculate how far horizontally it is from the center of motion after $0\mathrm{.}4$T$,$ making clear whether it is to the left or right from the center of motion, given that T $=1\mathrm{.}2$ s and d $=4\mathrm{.}2$ cm$.(1\mathrm{.}7$ cm$)$

$($iii$)$ Calculate the kinetic energy per unit mass when the ball bearing is at the center of motion.