So I've been having trouble with an exercise involving an inclined plane.
As shown in the figure, the inclined plane is being accelerated by the force F and, between the inclined plane and the box, there is a static friction μs.
It's asked for the maximum system acceleration possible in which there will be no movement between the inclined plane and the box, the force F to "produce" this acceleration and the Normal force done by the inclined plane over the box.
The think is, I don't know if the free body diagram for the box and my answers are correct.
I'll let my attempt below.
$$
\sum F_y~=~0\\\\N\cdot cos(\theta)-m\cdot g+\mu\cdot N\cdot sen(\theta)~=~0\\\\\boxed{m\cdot g~=~N\cdot \left(cos(\theta)+\mu\cdot sen(\theta)\right)}\\\\\\\sum F_x~=~0\\\\-m\cdot a+N\cdot sen(\theta)+\mu\cdot N\cdot cos(\theta)~=~0\\\\\boxed{m\cdot a~=~N\cdot \left(sen(\theta)+\mu\cdot cos(\theta)\right)}
$$
.
$$
\dfrac{m\cdot a}{m\cdot g}~=~\dfrac{N\cdot \left(sen(\theta)+\mu\cdot cos(\theta)\right)}{N\cdot \left(cos(\theta)+\mu\cdot sen(\theta)\right)}\\\\\dfrac{a}{g}~=~\dfrac{sen(\theta)+\mu\cdot cos(\theta)}{cos(\theta)+\mu\cdot sen(\theta)}\\\\\boxed{a~=~g\cdot\dfrac{sen(\theta)+\mu\cdot cos(\theta)}{cos(\theta)+\mu\cdot sen(\theta)}}
$$
.
$$
F~=~(m+M)\cdot a\\\\\boxed{F~=~(m+M)\cdot g\cdot \dfrac{sen(\theta)+\mu\cdot cos(\theta)}{cos(\theta)+\mu\cdot sen(\theta)}}
$$
.
$$
\boxed{N~=~\dfrac{m\cdot g}{cos(\theta)+\mu\cdot sen(\theta)}}
$$
