Axes: Choose along the incline (x') and perpendicular to incline (y').

Force components on incline at angle theta:

• Along incline (down): mg*sin(theta)
• Perpendicular to incline (into surface): mg*cos(theta)
• Normal force: N = mg*cos(theta) (for no perpendicular acceleration)

Case 1: No friction, no applied force: a = g*sin(theta) down the incline

Case 2: With friction, block sliding down: a = g*sin(theta) - $mu_k$gcos(theta) = g(sin(theta) - $mu_k$*cos(theta))

Case 3: With friction, block pushed up: a = g*sin(theta) + $mu_k$gcos(theta) = g(sin(theta) + $mu_k$*cos(theta)) (deceleration)

Case 4: Minimum force to push up the incline: $F_{min}$ = mg(sin(theta) + $mu_s$*cos(theta))

Case 5: Minimum force to prevent sliding down: $F_{min}$ = mg(sin(theta) - $mu_s$*cos(theta)) — only if sin(theta) > $mu_s$*cos(theta)