Combine two or multiple force vectors to find resultant magnitude, direction, x-y components, equilibrant, and optional acceleration with force-unit conversion.

Combine two forces in a plane.

Force 1 points along +x. Force 2 is counterclockwise from force 1.

Optional settings

Blank or zero omits acceleration. Enter mass only if these are all external forces on the object.

Resultant Force Formula

The following equation is used to calculate the resultant force acting on an object.

Fxy = Sqrt ((Fx)² + (Fy)²)
A = atan2(Fy, Fx)

Formula source: OpenStax University Physics, Volume 1 — Algebra of Vectors

  • Where Fxy is the magnitude of the resultant force
  • Fx is the sum of the x-components of forces
  • Fy is the sum of the y-components of forces
  • A is the resultant angle measured counterclockwise from +x, normalized to 0° ≤ A < 360°. It is undefined for a zero resultant.

For example, if there’s a force of 10 newtons acting in the positive x-direction and a force of 5 newtons acting in the -x direction, the result is 10 + (-5) = 5 newtons in the positive x-direction.

Resultant Force Definition

A resultant force is also known as the net force acting on an object. It’s the sum of all forces and their directions.

In global-angle mode, choose two to five force vectors and complete the fields shown. Zero magnitudes are allowed. The other modes use two forces with an included angle or two signed x-y component pairs.

Resultant Force Example

How to calculate resultant force?

Example #1:

The first step in solving for a resultant force is determining the forces and angles acting on the object. The following forces and angles will be used for this problem.

  1. 20 N @ 5 degrees
  2. 10 N @ 62 degrees
  3. 5 N @ 65 degrees
  4. 12 N @ 27 degrees
  5. 6 N @ 30 degrees

The next step is to calculate the X and Y components of each force. For the X-component this is done by multiplying the force by the cosine of the angle. For the Y-component this is done by multiplying the force by the sine of the angle.

Doing this yields the following X and Y components of force.

  1. X = 19.92 N, Y = 1.743 N
  2. X = 4.694 N, Y = 8.829 N
  3. X = 2.113 N, Y = 4.531 N
  4. X = 10.692 N, Y = 5.447 N
  5. X ≈ 5.196 N, Y ≈ 3.000 N

Next, sum the unrounded x-components and y-components. The totals are approximately 42.620 N and 23.552 N, respectively.

Using the unrounded totals, F = √(Fx² + Fy²) gives a resultant magnitude of approximately 48.695 N.

Finally, atan2(Fy, Fx) gives an angle of approximately 28.93° counterclockwise from +x.

Example #2:

This next example will involve only 3 different forces and angles, but some of the forces will have angles greater than 90 degrees. Those are as follows:

  1. 20 N @ 110 degrees
  2. 30 N @ 34 degrees
  3. 40 N @ 140 degrees

As done previously, first we calculate the x and y components. These are found to be:

  1. X = -6.840 N, Y = 18.793 N
  2. X = 24.871 N, Y = 16.775 N
  3. X = -30.641 N, Y = 25.711 N

The sums of the unrounded components are approximately −12.611 N in x and 61.281 N in y.

Using the unrounded component sums, the magnitude is approximately 62.565 N.

Finally, we calculate the angle using the y and x components of force. This is calculated to be 101.63 degrees.

Resultant in Physics

In physics, a resultant is the equivalent combination of the vectors of some property. This can be anywhere from velocity, to force, to acceleration. The only requirement is that the property being analyzed can be displayed as a vector.

This calculator applies force-unit conversions and labels its answers as force. Use it for forces; other vector quantities require their own units and interpretation.

The method for calculating the resultant, whether it be force or velocity, is the same.

Resultant Properties

For an object of positive mass in the classical model, zero net external force means zero translational acceleration. Rotational equilibrium also requires zero net torque, which this calculator does not determine.

It will stay still, or keep moving in the same direction and at the same speed as it was previously.

A resultant-force magnitude is nonnegative. It is zero when all force components cancel.

For a nonzero resultant, the displayed direction is normalized to 0° ≤ A < 360°, counterclockwise from +x. A zero resultant has no direction. Individual components may be negative, such as −10 N along x.

Units of Resultant Force

The SI standard units for resultant forces are newtons (N).

The English standard units are pound-force (lb-f).