The block of M$=5$kg$,$ modeled as a particle, rests on the inclined surface that makes an angle of $\mathrm{\backslash t}$heta with the horizontal ground. The static friction coefficient between the block and the surface is mu s$=1/3.$ A force of P is horizontally applied on the block. Take g$=10$m$/$s for convenience.

Q$1.$ Draw a free$-$body diagram $($FBD$)$ of the block.

Q$2.$ Write down Cartesian vector expressions for all forces shown on your FBD with respect to the specified coordinate system shown.

Q$5.$ Find the range of P needed to keep the block to rest on the surface without sliding down or pushing up 

Q$4.$ Find the critical value of P such that the static friction force between the block and the surface is zero.

$($Hint: The block could slide down for a significantly heavy block or be push up if P is too big$)$ 

Q$3.$ Write down the scalar component form of the equilibrium equations in the x$-$ and y$-$ directions.