Net Force Calculator

Forces are vectors: they have both magnitude and direction. To find the net (resultant) force on an object, you decompose each force into horizontal (x) and vertical (y) components, sum them separately, then recombine.

For a force F at angle θ (measured from the positive x-axis, counterclockwise):

Fx = F × cos(θ) Fy = F × sin(θ)

Sum all components:

Σ Fx = F1x + F2x + F3x Σ Fy = F1y + F2y + F3y

Net force magnitude: F_net = √(ΣFx² + ΣFy²) Direction: φ = arctan(ΣFy / ΣFx), adjusted for quadrant

Angle convention used here: 0° = right (east), 90° = up (north), 180° = left (west), 270° = down (south). Angles are measured counterclockwise from east.

If the net force is zero, the object is in equilibrium, and Newton’s first law says it stays at rest or continues in uniform motion.

Example: three forces of 10 N at 0°, 8 N at 90°, and 6 N at 210°. Fx = 10 + 0 + 6 cos(210°) = 10 − 5.196 = 4.804 Fy = 0 + 8 + 6 sin(210°) = 8 − 3 = 5 F_net = √(4.804² + 5²) = √(23.08 + 25) = √48.08 ≈ 6.93 N

Forces F2 and F3 are optional. Leave them blank or at 0 N if you only have one or two.

Equilibrium is not the same as “no forces”

A net force of zero means the forces present cancel each other out, which is a real and useful result: a book on a table, a lamp on a chain, a car at steady speed on a level road. All of those have forces acting and a net force of zero. An empty form is a different situation entirely, so the calculator asks for input rather than reporting equilibrium.

Worth noting that equilibrium says nothing about whether the object is moving. Newton’s first law lumps “at rest” and “moving at constant velocity” together on purpose: neither requires a force to maintain. A puck sliding on frictionless ice and a puck sitting still are in the same mechanical state.

Sign conventions and the quadrant trap

The direction comes back from a two-argument arctangent, which is why the calculator can tell 135° from 315° even though the ratio ΣFy/ΣFx is identical for both. A single-argument arctan cannot, and reporting the wrong quadrant is the classic error when people work this out by hand with a calculator’s atan key. If you are checking the arithmetic yourself, sketch the components first and confirm the answer lands in the quadrant your sketch shows.

Three forces is enough for most textbook problems, but the method does not care how many you have. Decompose everything into x and y, add the columns, recombine. A hundred forces works exactly the same way.