Net Force Calculator: Resultant of Several Forces (Magnitude, Direction and Acceleration)

How it works

Fx = F·cos θ, Fy = F·sin θ for each force · ΣFx = F₁x + F₂x + …, ΣFy = F₁y + F₂y + … · |F_net| = √(ΣFx² + ΣFy²) · θ = atan2(ΣFy, ΣFx) · a = F_net ÷ m

The net force is the single force that has the same effect as all the forces acting on an object together. Forces are vectors, so you cannot just add their sizes: split each force into x and y components with cosine and sine, add all the x parts and all the y parts, then rebuild one vector. Its length, √(ΣFx² + ΣFy²), is the size of the net force, and atan2(ΣFy, ΣFx) gives its direction in the correct quadrant. Enter forces as a magnitude and an angle (counterclockwise from the +x axis, or as compass bearings under More options) or directly as x and y components. For forces along one line, type one number per line with a minus sign for the opposite direction. The calculator shows every component, the net force, the balancing force (equal in size and opposite in direction, which would put the object in equilibrium) and, if you give the mass, the acceleration from Newton’s second law. It adds the forces you list; it does not work out weight, normal force or friction for you. For those, use the friction force or inclined plane calculators, and for a single force from mass and acceleration use the F = ma force calculator.

Worked example

Three forces act on a crate: 50 N at 0°, 30 N at 90° and 20 N at 210°. The x components are 50, 0 and 20 × cos 210° = −17.32 N, so ΣFx = 32.68 N. The y components are 0, 30 and 20 × sin 210° = −10 N, so ΣFy = 20 N. The net force is √(32.68² + 20²) = 38.31 N at atan2(20, 32.68) = 31.5° above the +x axis. A balancing force of 38.31 N at 211.5° would hold the crate still, and for a 10 kg crate the acceleration would be 3.831 m/s².

Sources: OpenStax University Physics Vol. 1, 5.1 Forces (net force of 30.0 N and 40.0 N; component example)

Frequently asked questions