Answer all parts with full work showing.

Two boxes with masses $m_1$ and $m_2$ lean on each other on an inclined plane, with box $m_2$ closer to the floor than box $m_1.$ The boxes are made of different materials such that the coefficient of kinetic friction of box $1$ is ${}_1$ and the coefficient of kinetic friction of box $2$ is ${}_2($with The height of the inclined plane is $h$ and the length of the incline in $l.$ Initially you are holding box $m_2$ such that neither box is moving.

a$.$ In terms of the given quantities write an equation for the angle $$ that the base of the incline makes with the horizontal.

b$.$ Draw a co$-$ordinate system that has one axis parallel to the inclined plane $($call it $x^{\text{'}})$ and one axis perpendicular to the inclined plane $($call it $y^{\text{'}}),$ and then determine the coefficients of the gravitational acceleration with respect to the $x^{\text{'}}$ and $y^{\text{'}}$ directions of this new co$-$ordinate system.

c$.$ Draw a free body diagram of the system $($box $1$ and box $2$ separately$)$ with respect to the $x^{\text{'}}$ and $y^{\text{'}}$ co$-$ordinate system from part b$).$ Label ALL forces.

d$.$ If you now let go of box $m_2$ and the boxes begin sliding together $($i$.$e$.$ in contact$)$ down the incline, write the equations for the acceleration $($derived from Newton's $2^{nd}$ Law$)$ of box $1$ with respect to the $x^{\text{'}}$ and $y^{\text{'}}$ axes $($Note: I want $2$ equations, one for the acceleration in the $x^{\text{'}}$ direction, and one for the acceleration along the $y^{\text{'}}$ direction$).$

e$.$ If you now let go of box $m_2$ and the boxes begin sliding together down the incline together $($i$.$e$.$ in contact$),$ write the equations for the acceleration $($derived from Newton's $2^{nd}$ Law$)$ of box $2$ with respect to the $x^{\text{'}}$ and $y^{\text{'}}$ axes $($Note: I want $2$ equations, one for the acceleration along the $x^{\text{'}}$ direction, and one for the acceleration along the $y^{\text{'}}$ direction$).$

f$.$ Based on your above equations, derive an equation for the acceleration of box $m_2$ along the $x^{\text{'}}$ axis that depends only on $,{}_1,{}_2,m_1,m_2$ and $g.$

g$.$ Derive an equation for the magnitude of the contact force between the $2$ boxes in terms of the parameters $,{}_1,{}_2,m_1,m_2$ and $g.$

h$.$ At what angle will the boxes have an acceleration of $0$ along the incline, express your answer in terms of ${}_1,{}_2,m_1,m_2$ and $g?$