Please help with clear explanation.

Rigid$-$Body Dynamics Project

Instructions

There is a disk of radius r and mass m rotating inside half of a cylinder of radius R and mass M $.$

$M=578$ grams

$m=162$ grams

$R=15\mathrm{.}625cm$

$r=3\mathrm{.}8cm$

Figure $1-$A disk of

radius $r$ rolls inside a

half$-$cylinder of radius

capital$-$R that rolls on

a horizontal surface.

Determine the center of mass and mass moment of inertia of the half of a cylinder

ANALYTICALLY. Show your work! This is a Statics problem.

Determine the EOM of just the half$-$cylinder if the disk does not exist. Plot the angular

position for $10$ seconds given an initial condition of $=\frac{}{9}$rad and ${}^{}=0ra\frac{d}{sec}.$ Using the

period of oscillations, what is the natural frequency of this system?

Determine the EOM of the disk if the cylinder is fixed. Plot the angular position for $10$

seconds given an initial condition of $=\frac{}{9}$rad and ${}^{}=0ra\frac{d}{sec}.$ Using the period of

oscillations, what is the natural frequency of this system?

Determine the EOMs for the system, where both the cylinder and the disk move.

Integrate the full EOMs for the system using various initial conditions to show the different

styles of motion that the system exhibits.

a$.$ Initial conditions: $=0$rad,${}^{}=0ra\frac{d}{sec,}=\frac{}{18}$rad,${}^{}=0ra\frac{d}{sec}$

Plot the response for $6$ seconds.

b$.$ Initial conditions: $=\frac{}{18}$rad,${}^{}=0ra\frac{d}{sec,}=\frac{}{9}$rad,${}^{}=0ra\frac{d}{sec}$

Plot the response for $6$ seconds.

c$.$ Initial conditions: $=\frac{}{9}$rad,${}^{}=0ra\frac{d}{sec,}=-\frac{}{9}$rad,${}^{}=0ra\frac{d}{sec}$

Plot the response for $6$ seconds.