Problem $13$: A uniform horizontal plank which has a mass $m_p=5\mathrm{.}0kg$ and a length $L=4\mathrm{.}0m$ is suspended by a hinge on its right end and a rope on its left end. A weight with a mass of $m_w=10\mathrm{.}0kg$ rests at the midpoint of the plank. At $t=0,$ the rope is cut, allowing the plank to rotate about the hinge.

a$)$ What is the rotational inertia I of the plank$-$weight system about the plank's hinge?

Hint: The moment of inertia of a thin rod of mass $m$ and length $L$ about an axis through the rod's center and perpendicular to its length is equal to $\frac{m{L}^2}{12}$

b$)$ What is the magnitude of the torque $$ and its direction $($positive or negative$)$ acting on the system due to the gravitational force?

c$)$ What is the magnitude of the angular acceleration $$ of the plank about the hinge just after $t=0?$

d$)$ What is the magnitude of the downward accel$$tation $a$ of the weight just after $t=0?$

e$)$ What is the magnitude of the normal force $F_N$ acting on the weight just after $t=0?$

Thin rod about axis through center perpendicular to length