Inclined Plane Calculator
The inclined plane is one of the classic problems in physics and one of the most useful for building intuition about forces. When an object rests on a slope, gravity still acts straight down, but it is convenient to resolve it into two components: one running along the slope and one pressing into the slope. The component along the slope, equal to the weight times the sine of the angle, is what tends to pull the object down the ramp. The component into the slope, equal to the weight times the cosine of the angle, is what the surface resists with the normal force. If there is friction, it opposes the downslope motion with a force equal to the kinetic coefficient of friction times the normal force. The net force down the slope is therefore the parallel gravity component minus the friction force. Divide by the mass and you get the acceleration. If friction is large enough to equal or exceed the gravity component, the net force is zero and the object stays put. This calculator resolves all of these forces at once. Enter the slope angle in degrees, the mass of the object, and the coefficient of friction, and it returns the force parallel to the slope, the normal force, the friction force, the net force, and the resulting acceleration. The default example uses a 30 degree slope, a 10 kg mass, and a coefficient of friction of 0.2. The parallel force is 49.05 N, friction is 16.99 N, the net force is 32.06 N, and the acceleration is 3.21 m/s2. Results update as you type. Use this for physics homework, for understanding vehicles on gradients, for conveyor and chute design, or for any ramp-related engineering problem. g is taken as 9.81 m/s2.
Forces in newtons using g = 9.81 m/s². If friction holds the object still, acceleration is shown as 0. Rounded for display.
How it works
With weight W = mg: parallel component F_par = W sin(θ); normal force N = W cos(θ); friction force F_fric = μ N. Net force down slope: F_net = max(0, F_par - F_fric). Acceleration: a = F_net / m = g(sinθ - μ cosθ) when the object slides; 0 if friction holds it. g = 9.81 m/s².
Worked example
Slope θ = 30 deg, mass m = 10 kg, μ = 0.2. Weight = 10 times 9.81 = 98.1 N. F parallel = 98.1 times sin(30) = 98.1 times 0.5 = 49.05 N. Normal force = 98.1 times cos(30) = 98.1 times 0.866 = 84.96 N. Friction = 0.2 times 84.96 = 16.99 N. Net force = 49.05 - 16.99 = 32.06 N. Acceleration = 32.06 / 10 = 3.21 m/s². These match the default values pre-filled above.
Related calculators
- Friction Calculator: static and kinetic friction on a flat surface.
- Force Calculator: net force from mass and acceleration.
- Average Velocity Calculator: velocity from displacement and time after the object slides.
- Displacement Calculator: distance travelled on the slope from acceleration and time.