Question: [Physics] a moving inclined plane with block

Hello everybody,

I'm trying to solve for a challenging problem : a moving inclined plane with a block

I want to solve for the acceleration components for the block and the plane and the normal force acting on the block.

Let O=(0,0) be an external origin.

Let h be the upper left height of the inclined plane.

Let x₁ be the x-position of the center of gravity of the inclined plane.

Let x₂ be the x-postion of the center of gravity of the block.

Let y be the y-position of the center of gravity of the block.

Let m₁ be the mass of the plane. Let m₂ be the mass of the block.

Let   be the coeffiction of kinetic friction between the bottom of the inclined plane and the level surface.

Let  be the coeffiction of kinetic friction between the block and the upper surface of the inclined plane.

Let  be the angle of the plane with the horizontal.

Let F_p a force applied to the inclined plane.

With those defined variables, I make two separable free body diagrams for the block and for the inclined plane, indicating all of the external forces acting on each. It then comes those two vectorial equations :

Block : m₂a₂=W_{weight of block}+F_{plan acting on block}+F_{friction from plan to block}+N_{normal from plan to block}

Plane : m₁a₁=W_{weight of plane}+F_{pushing force}+F_{block acting on plane}+F_{friction from level to plan}+N_{normal from level to plane}+F_{friction from block to plane}+N_{normal from block to plane}

I am quite not sure whether I should include the F_{friction from block to plane} and the N_{normal from block to plane} into the plane's acceleration calculation. Am I right ?

I notice that from the geometry of the figure, I can write down the relation : tan()=(h-y)/(x₂-x₁)

This implies the relation : -a_2y=tan()(a_2x-a_1x) (equation 1)

Writing down the equations for the x- and y- components of the accelerations of the block and of the plane , this yields :

( equation 2) : m₂a_2x=m₁ sqrt(a_1x²+a_1y²) cos() -     N₁ sin() +N₁ cos() 

(equation 3) : m₂a_2y= m₂g+m₁ sqrt(a_1x²+a_1y²) sin() +   N₁ cos() +N₁ sin() 

(equation 4) : m₁a_1x=Fp - m₂ sqrt(a_2x²+a_2y²)  sin() +   N₁ cos() - N₁ cos() 

(equation 5) : m₁a_1y=-m₁g  - m₂ sqrt(a_2x²+a_2y²)  cos() -  N₁ + N₁ -   N₁ sin() -   N₁ cos()

Since N₁=m₁g,  equation 5 becomes : m₁a_1y= - m₂ sqrt(a_2x²+a_2y²)  cos()   -   N₁ sin() -   N₁ cos()

I am confused at this stage because a_1y=0, that is to say, the plane remains at the ground level surface.

Where am I wrong ? Does this comes from my previous question ?

I want to solve this problem with Maple and plot the solutions. Thank you for any answer !