Homework H$\mathrm{.}6.$D

Given: A thin, homogeneous bar OA$,$ having a mass of $M$ and length $L,$ is supported by a smooth pin joint at end O$.$ A bullet B $($mass $m)$ strikes end A of the stationary bar with a speed of $v_0,$ with the bullet sticking immediately after impact. Let $(t)$ represent the angular motion of bar OA about pin O during the subsequent motion of the system. Assume that the motion of the bar is small, such that and that the bullet can be modeled as a particle.

Find: For this problem:

$($a$)$ Determine the angular speed ${}^{}(0)$ of the bar immediately after the bullet has stuck to the bar;

$($b$)$ Determine the dynamical equation of motion $($EOM$)$ of the bar$/$bullet system in terms of $$ that will describe its motion immediately after the bullet has stuck to the bar;

$($c$)$ The EOM that you found above will be "nonlinear" in the dependent variable $$ due to the presence of the $sin$ term. Note that for small angles $,$ we can approximate $sin$ by $$; that is$,sin$~~$.$ Make this replacement in your EOM to produce its "linearized" form valid for small angles of oscillations.

$($d$)$ Determine undamped natural frequency ${}_n$ for the system; and,

$($e$)$ Determine an expression for $(t)$ describing the motion of the system for $t>0.$