Problem $4$: $(20$ points$)$

As the figure shown below, the slender rod has a mass of $1$ kg and length of $50$ cm is attached by the pin $O.$ The sphere has a mass of $2$ kg and radius of $2$ cm $.$ The sphere is attached at the end of the slender rod. The body is released from the rest in the horizontal position, then it reaches the vertical position. Assume gravity $g=9\mathrm{.}81\frac{m}{s^2}.$ Show all work.

a$)$ Draw the appropriate free body and kinetic diagram$($s$)$ before and after release. Determine the datum line. Please show it visually in the chegg answer.

b$)$ Write the appropriate energy conservation equation based on the free body and kinetic diagram$($s$)$ written above. $(2$ points$)$

c$)$ Compute the kinetic and gravitational energy at horizontal position. $(4$ points$)$

d$)$ Compute the kinetic and gravitational energy at vertical position. $(4$ points$)$

e$)$ Determine the total mass moment of inertia, $I_D,$ about pin, $O.(4$ points$)$

f$)$ Determine the magnitude of its angular velocity at vertical position, ${}_0.(2$ points$)$