Block sliding down accelerated inclined (need inertial frame approach)

[PhysicsDaoist]

Homework Statement

A block of mass "m" is placed on an incline of angle [tex]\theta[/tex] and mass "M", which is placed on a horizontal surface (the ground) and release from rest. Both the block and incline start accelerate. All surfaces are frictionless. Find the acceleration of the block with respect to the incline.

2. The attempt at a solution
I have no problem solving this using non-inertial frame and pseudo-force (inertial force).
Separately, I did FBD on the block (non-inertial frame) and FBD on the incline (inertial reference from the ground). I've solved the equations and found Incline acceleration, normal force between block and incline and the relative acceleration of block to inclined.

a_m = mg cos(theta) sin(theta) / (M + m sin^2 (theta))

f_N = Mmg cos(theta) / (M + m sin^2 (theta))

a_mM = (M+m)/(M+m sin^2 (theta)) * g sin (theta)

I was told that one always solve these from an inertial frame and the results will be the same.
However, I failed to solve this referencing from the ground at an inertial frame.
I started with the followings:

For small block m - along x: -f_N sin (theta) = - m a_x
along y: -mg + f_N cos(theta) = - m a_y

(Intuitively, I roughly know the direction of the small block of going down to the left)

For incline M along x: f_N sin(theta) = M a_M
along y: -Mg - f_N cos(theta) + F_N = 0
(F_N is the normal force from the ground to the incline)

I am stucked not able to find other relations to solve a_M and the rest.
Is this possible? without the use of inertial force and be solved under an inertial frame?

I also attempted to start the problem with using the Center-of-mass of this system (block+incline) since the only force to the system (as a whole) is (M+m)g and work from that with no luck.

 

[anathema]

Thank you for posting your question. It is always important to consider different reference frames when solving a problem in physics. In this case, you are correct that solving the problem in an inertial frame would give the same results as solving it in a non-inertial frame with the addition of pseudo-forces.

When considering the problem in an inertial frame, it is important to remember that all forces acting on the system must be taken into account. In this case, the forces acting on the system are gravity, the normal force from the incline, and the normal force from the ground.

Using the free body diagrams that you have already created, we can set up the equations of motion for the system in the x and y directions.

For the block m:
- mg sin(theta) + f_N = m*a_x
- mg cos(theta) - N = m*a_y

For the incline M:
- M*g + N = M*a_M

We also know that the acceleration of the block with respect to the incline is given by:
a_m = a_x - a_M

From these equations, we can solve for the acceleration of the block with respect to the incline, a_m, as well as the normal force between the block and the incline, f_N. We can then use these values to find the acceleration of the block with respect to the ground, a_x, and the acceleration of the incline, a_M.

I hope this helps to clarify the problem and how to approach it from an inertial frame. Let me know if you have any further questions.