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PhysicsDaoist

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## 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.

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