Question $3$

$[33$ marks$]$

For an aircraft touch$-$down problem, wheels on a flat plane start from slipping $($no rolling$),$ with $\backslash ($ v$\_\{0\}=23\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$ Other wheel initial conditions are shown in Fig. $3\mathrm{.}1.$

$($a$)$ Find the time taken for the wheel to stop slipping and the distance travelled before it stops slipping. Assume a constant acceleration and a cylinder wheel with uniform density.

$[20$ marks$]$

$($b$)$ If the same wheel has an initial rotational speed of $\backslash ($ w $\backslash )$ before being freely released to the ground, determine the time that it takes for the cylinder to move from initial touch$-$down slipping to rolling. You are not required to calculate the numerical values.

$[13$ marks$]$

Fig. $3\mathrm{.}1$: Aircraft touch$-$down initial conditions