Inclined Planes
Tsokos Ch. 3.4
Overview
For an object on a slope of angle θ, resolve the weight W = mg into two components: along the slope: W sinθ (causes sliding), perpendicular to slope: W cosθ (determines N). The normal force N = mg cosθ (if no other perpendicular forces). Friction force = μ_k · mg cosθ. Net force along slope = mg sinθ − f.
Setting up incline problems
For a block on an incline at angle θ: set up a coordinate system along the slope (x-axis parallel to slope, y-axis perpendicular). Forces: Weight components: W_x = mg sin θ (down the slope), W_y = mg cos θ (into slope). Normal force: N = mg cos θ (since a_y = 0 perpendicular to slope). Friction (if rough): f = μN = μmg cos θ (up the slope if block slides/tends to slide down). Apply F_net = ma along the slope.
Worked Examples
Common Mistakes
- ⚠Using the full weight mg instead of its component mg sin θ along the slope
- ⚠Forgetting that N = mg cos θ (not mg) on an incline