Step 1: FBD
Draw individual free-body diagrams for the plate and cart.
Step 2: Kinetics - Newton/Euler
Write down the Newton/Euler equations for the plate and the cart. Note that there are no fixed points observed for the plate; therefore, you need to use a moment equation of the plate about the plate's center of mass.
Step 3: Kinematics
Use the rigid body kinematics equations to relate the acceleration of the cart to the acceleration of the plate's center of mass through the angular acceleration of the plate.
Step 4: Solve
Solve your equations from Steps 3 and 4 for the angular acceleration of the plate and the acceleration of the cart.