Inclined Plane Calculator

Forces on an inclined plane

Three forces act on an object on a slope: its weight (straight down), the normal force (perpendicular to the surface) and friction (along the surface, opposing motion). To make the problem easy, the weight is split into two components:

• Along the plane: m × g × sin θ (pulls the object downhill)
• Perpendicular to the plane: m × g × cos θ (equal to the normal force)

Acceleration

On a frictionless plane the acceleration is a = g × sin θ and does not depend on mass. With friction, a sliding object accelerates at a = g × (sin θ − μ × cos θ).

Will it slide?

The object slides only if the component along the plane exceeds the maximum friction. That condition simplifies to tan θ > μ. The angle at which sliding starts, θ = arctan μ, does not depend on mass. This is how the coefficient of friction is measured: place the object on a board and tilt the board slowly.

Example

A 10 kg crate sits on a 30 degree plane with μ = 0.2 and g = 10 m/s². The weight is 100 N, the component along the plane 50 N, the normal force 86.6 N and friction 0.2 × 86.6 = 17.3 N. The net force is 32.7 N and the acceleration 3.27 m/s². Pulling the crate uphill at constant speed takes 50 + 17.3 = 67.3 N.

Limits

The tool is for a sliding object. For rolling objects (a ball, a cylinder, a wheel) part of the energy goes into rotation and the acceleration is smaller. The static and kinetic coefficients are taken as equal. The pulling force is assumed parallel to the plane: pulling at an angle changes the normal force.