Inclines

Inclines are surfaces that rise away from the horizontal, characterized by an angle or a grade. They occur in nature, such as hills and slopes, and in engineered forms like ramps, roads, stairs, and slides. An incline may be straight or curved, and its steepness influences how it is used and perceived.

Measurement: The grade or slope is rise over run, often expressed as a percentage: grade = (rise/run) × 100. The incline angle θ relates to the same geometry by tan θ = rise/run. The hypotenuse length L of a right triangle with rise h and run r is L = sqrt(h^2 + r^2).

Physics: A body on an incline experiences gravity decomposed into a parallel component mg sin θ and a perpendicular component mg cos θ. Friction depends on the normal force; the work required to move the body up the incline equals m g h, regardless of the path, in the absence of losses.

Applications and types: Natural inclines include hills and escarpments; human-made inclines include vehicle ramps, pedestrian ramps, roads with grades, ski slopes, and escalators. Inclined planes are one of the six classical simple machines; their mechanical advantage equals the length of the incline divided by the vertical rise (ideal, frictionless case).

Design and safety: Slope limits for accessibility and safety vary by region. For wheelchair access in many standards, a common maximum slope is about 1:12 (roughly 8%). Designers also consider handrails, landings, drainage, and traction to prevent slipping on wet or icy surfaces.