Force Decomposition: Weight (mg) splits into parallel component F∥ = mg·sin(θ) pulling down the slope, and perpendicular component F⊥ = mg·cos(θ) pushing into the surface. As angle increases, F∥ increases (more sliding force) and F⊥ decreases (less normal force).
Normal Force: N = mg·cos(θ) is the perpendicular contact force from the surface. It equals F⊥ when no other vertical forces act. Normal force decreases with steeper angles, reaching zero at 90° (vertical).
Friction: Opposes motion with f = μN = μmg·cos(θ). Static friction (μ_s) keeps objects at rest. Kinetic friction (μ_k, smaller) acts on moving objects. Critical angle where sliding starts: θ_c = arctan(μ_s).
Acceleration: Without friction: a = g·sin(θ). With friction: a = g(sin(θ) - μ·cos(θ)). If μ·cos(θ) > sin(θ), net force is zero and object doesn't slide.
Mechanical Advantage: MA = 1/sin(θ) = L/h (ramp length/height). Gentler slopes need less force but longer distance. Work is conserved: F_effort × L = mg × h.
Applications: Wheelchair ramps (ADA max 4.76° or 1:12), road grades, loading ramps, ski slopes, conveyor belts, roof pitch, landslide stability analysis in geotechnical engineering.