Discussion and hints:

The derivation of the dynamical equation of motion (EOM) for a system is a straight-forward application of what we have learned from Chapter 5 in using the Newton-Euler equations. The goal in deriving the EOM is to end up with a single differential equation in terms of a single dependent variable that describes the motion of the system. Here in this problem, we want our EOM to be in terms of x(t).

Recall the following four-step plan outline in the lecture book and discussed in lecture:

Step 1: FBDs
Draw a FBD of the disk. Define a rotation coordinate for the disk.

Step 2: Kinetics (Newton/Euler)
Write down the Newton/Euler equations for the disk.

Step 3: Kinematics
Use the no-slip condition between the disk and the ramp to relate x to the rotational coordinate that you chose above.

Step 4: EOM
Combine your Newton/Euler equations along with your kinematics to arrive at a single differential equation in terms of the dependent variable x.

Any questions?