Question $3(24$ Marks Total$)$

Small parts of mass $1\mathrm{.}5$ kg $,$ moving at a speed of $\backslash (2\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ on a production line, slide down a fixed, smooth ramp and are brought to rest by a spring of stiffness, $\backslash ($ k $\backslash ),$ as illustrated in Figure Q$3.$ The maximum allowable deceleration to avoid damage to the components is $\backslash (95\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}^\{2\}\backslash ).$

$($a$)$ Determine the maximum spring stiffness, $\backslash ($ k $\backslash ),$ to ensure that the allowable deceleration is not exceeded. Give the result in N$/$m to three or four significant figures.

$[12$ marks$]$

$($b$)$ Determine the deflection of the spring corresponding to the value of $\backslash ($ k $\backslash )$ found in part $($a$).$ Give the result in mm to three or four significant figures.

$[12$ marks$]$