Problem $4.$ The experiment shown in the figure is

used to measure the mass moment of inertia of a

tennis racket. A racket of mass $m$ is hung and swings

about point $O.$ The center of mass $G$ of the racket is at

distance $d$ from $O.$ The angle of the racket is $.$

a$)$ Determine the DoF and derive the EoM for $$ using

FBD and FLD$.($Hint: use moment equation about $O.)$

b$)$ Linearize the EoM about the equilibrium ${}^{*}=0.$

Solve the linearized EoM with $(0)={}_0$ and

${}^{}(0)=0$ analytically.

c$)$ The duration of $10$ periods of small oscillations

$($swings$)$ is measured as $10T=20\mathrm{.}5s.$ Considering

$m=0\mathrm{.}3kg,g=9\mathrm{.}81\frac{m}{s^2},$ and $d=1m,$ determine

the mass moment of inertia $J_G$ of the racket about $G.$

d$)$ Simulate the motion in Matlab by solving the

nonlinear EoM numerically for ${}_0=\frac{}{6}.$ Plot the

numerical solution $(t)$ over tin$[0,10]$ s$.$ Plot the

analytical solution $(t)$ of the linearized EoM in the

same graph. Submit the figure and your codes.