Resolve gravity into components parallel and perpendicular to an incline and find the acceleration of an object sliding on a ramp, with and without friction.
An object on a ramp needs the SAME component skills from week 3, applied to gravity itself. Instead of gravity (m*g, straight down) acting simply, it is tilted relative to the ramp's surface, so it splits into two useful pieces: a component PARALLEL to the ramp's surface, m*g*sin(theta), which pulls the object down the slope, and a component PERPENDICULAR to the surface, m*g*cos(theta), which the ramp's normal force must balance. Notice that on an incline, the normal force is NOT equal to the full weight anymore -- it only balances the perpendicular piece, m*g*cos(theta), which shrinks as the ramp gets steeper.
For a frictionless incline, the parallel component of gravity is the only force along the slope, so Newton's second law gives a wonderfully simple result: a = g*sin(theta) down the ramp, independent of mass. Add kinetic friction (for an object sliding down) and it opposes the motion, acting up the slope: a = g*(sin(theta) - mu_k*cos(theta)). As theta approaches 90 degrees (a vertical wall), this whole setup smoothly turns into ordinary free fall, since sin(90)=1 and cos(90)=0.
Mastery looks like: They correctly resolve gravity into components on an incline and compute the resulting acceleration, with and without kinetic friction.
Common stumbles: Using the full weight (m*g) as the normal force on an incline instead of only the perpendicular component; mixing up sin and cos between the parallel and perpendicular components.