Problem #$6(15$ points$)-$ A Falling Rod

The figure below shows a uniform rod of length $L$ that falls so that the bottom of the rod

slides without friction along a horizontal floor.

The falling rod with the angle $$ indicated

If $$ is the angle between the bottom of the rod and the floor, it can be shown using forces

and torques that $$ satisfies the ODE

$(1+3co{s}^2()){}^{}-3{}^{}{?}^{2}sin()cos()+6{}^{2}cos()=0$

where ${}^2=\frac{g}{L}.$ Assuming the initial conditions $(0)={}_0=\frac{}{3}$rad and ${}^{}(0)={}_0^{}=0$

and that $L=6\mathrm{.}9m,$ along with $g=9\mathrm{.}8\frac{m}{s^2}$ and conservation of energy, which leads to

${}^{}{?}^2=12{}^2(\frac{sin({}_0)-sin()}{1+3co{s}^2()})$

determine the time it takes for the entire rod to hit the ground in terms of an integral and

then evaluate this integral numerically using any software of your choosing.