Find the force needed for a body on an inclined plane to be static?

[Like Tony Stark]

Homework Statement

The picture shows a body of mass $m_1$ lying on an inclined surface of mass $m_2$. Suppose that all surfaces are frictionless. Determine the horizontal accelaration that sould be applied to $m_2$ to keep $m_1$ in equilibrium. Then, suppose that an acceleration twice bigger than the calculated previously is applied to $m_2$, what's the acceleration of $m_1$ with respect to Earth (inertial system) and with respect to $m_2$?

Relevant Equations

Newton's equations

Here, I have two doubts
1) if the surfaces are frictionless, then there's no force being applied on the $x' axis$ of $m_1$ except from the weight, so it should be sliding, shouldn't it? So, there's no force that I could apply to $m_2$ to keep $m_1$ in equilibrium since any force that I apply on $m_2$ will manifest in the normal force of $m_1$ and this is on the $y'$ axis of $m_1$.

2) And then I have some problems with "acceleration with respect to...". If I use Newton's equations and I solve for $a$, that acceleration is with respect to $m_2$ or with respect to Earth?

 

[PeroK]

Here, I have two doubts
1) if the surfaces are frictionless, then there's no force being applied on the $x' axis$ of $m_1$ except from the weight, so it should be sliding, shouldn't it? So, there's no force that I could apply to $m_2$ to keep $m_1$ in equilibrium since any force that I apply on $m_2$ will manifest in the normal force of $m_1$ and this is on the $y'$ axis of $m_1$.

Your doubts shouldn't stop you doing the calculations.

2) And then I have some problems with "acceleration with respect to...". If I use Newton's equations and I solve for $a$, that acceleration is with respect to $m_2$ or with respect to Earth?

Newton's laws apply in an inertial reference frame.