Formula and Method for the Inclined Plane Calculator
An inclined plane is one of the six classical simple machines: a flat, sloped surface used to raise a load with less force than lifting it straight up, at the cost of moving it a longer distance. This calculator resolves the load's weight into components along and perpendicular to the slope, adds friction, and reports the force needed to push the load up, the normal force pressing it into the surface, the plane's ideal mechanical advantage, and the height it gains over the given slope length.
Breaking the weight into components
Gravity pulls straight down on the load with force W = mg. On a slope tilted at angle θ from horizontal, that force splits into two perpendicular components: W sin θ acting parallel to the surface (pulling the load back down the slope) and W cos θ acting perpendicular to the surface (pressing the load into the ramp). The surface pushes back with an equal and opposite normal force, N = W cos θ, which is what keeps the load from sinking into the ramp. On a frictionless incline, the parallel component alone determines both the force needed to hold a load in place (F = W sin θ) and the acceleration of a released object down the slope (a = g sin θ) — never the full value of g, since only part of gravity acts along the surface.
Adding friction and the self-locking angle
Real ramps have friction. Kinetic friction opposes the load's motion with magnitude f = μN = μW cos θ, where μ is the coefficient of friction between the load and the surface. Pushing the load up the slope at a constant speed therefore takes F = W(sin θ + μ cos θ) — the friction force adds directly to the weight component doing the work of climbing. Reverse the sign for a load sliding down: friction subtracts from the driving component instead. When μ is at least tan θ, friction alone can supply enough opposing force to keep a stationary load from sliding — this is called a self-locking incline, and the angle where μ = tan θ is the "angle of repose" for that surface pair. Below that friction level, the load needs to be actively restrained or it accelerates back down the ramp.
Mechanical advantage and real-world ramps
The ideal (frictionless) mechanical advantage of an inclined plane is MA = 1 / sin θ, which equals the slope length divided by the height gained (L / h) — a direct consequence of the work-energy trade-off: spreading the same lift over a longer distance lowers the force needed at any instant. Loading ramps, wheelchair ramps, mountain roads, and even wedges and screws (inclined planes wrapped or driven) all use this trade-off. Accessibility codes such as the ADA cap ramp slope at roughly 1:12 (about 4.8°) specifically to keep the required push force low enough for a person in a wheelchair to manage unassisted.