A $50$ kg block is at rest on an inclined plane. The plane makes an angle $\backslash$theta with the horizontal. The coefficient

of static friction between the plane and the block is $$s $=0\mathrm{.}4,$ while the coefficient of kinetic frrction is $$k $=0\mathrm{.}2.$

$($a$)$ Sketch a free body diagram for the block. Label all of the forces!

$($b$)$ Show that the block will remain at rest if tan $\backslash$theta $<=$s$.$ There are several lines of algebra and trigonometry

here.

Assuming that tan $\backslash$theta $=$s for the remainder of this problem.

$($c$)$ A student give the block a gentle nudge down the incline. What is the resulting acceleration of the block?

$($d$)$ Another student grabs the block, and gives it a single shove up the incline with an initial speed of $10$ m$/$s$.$

Describe what the block does