A throwing dart is thrown at a target which is is free to pivot around a rotational axis. The dart is thrown at the target and collides in a perfectly inelastic collision, which sets the target in motion around the pivot. The geometry of the masses are as follows: $m_4=1\mathrm{.}27kg$ is an aluminum disk of radius $r_4=0\mathrm{.}15m$ which has a bearing at its center. This pivot point is shown as a dot in the figure. $m_3=0\mathrm{.}150kg$ is a uniform rod of length $l=1\mathrm{.}0m,$ and is attached to the ruler $20$ cm from its end. $m_5=0\mathrm{.}5kg$ is a solid cylinder of radius $r_5=0\mathrm{.}02m$ sitting on top of the ruler $l_5=-0\mathrm{.}15m$ from the pivot, and $m_2($assume massless$)$ is the target for the dart, located $l_2=0\mathrm{.}6m$ from the pivot. The dart is shown as $m_1=0\mathrm{.}10kg$ and collides directly with the center of the target $(0\mathrm{.}6$ m from the pivot$).$

a$)$ Calculate the angular momentum of the sy stem before the collision, assuming the dart is thrown at $v_{1i}=12\frac{m}{s}.$

b$)$ Calculate the moment of inertia of the rotating masses with the dart stuck into the target.

c$)$ Calculate the final angular velocity of the system after the collision.