Net Force Calculator

What the net force calculator does

When several forces act on the same object at once, what matters is their combined effect, the single force that would do the same job. This calculator finds that net force. You enter up to ten forces, each with its size and the direction it points, and it returns the resultant: its horizontal and vertical parts, its overall size, and the direction it acts in.

Because forces have direction, they cannot simply be added as plain numbers. Below is how they combine, the equations behind it, and a worked example.

How to use it

1. Enter each force with its magnitude and its angle, measured from the positive x-axis. A new row opens up as you go, up to ten forces.
2. Choose units for each force and angle, newtons or other force units, degrees or radians.
3. Press Calculate for the resultant's components, magnitude, and direction, or Reset to clear it.

Forces are vectors, so they add as arrows

A force has both a size and a direction, which makes it a vector. Two forces of 10 newtons each do not always combine into 20. Pulling the same way, they do. Pulling against each other, they cancel to nothing. Pulling at an angle, they make something in between. So you cannot add the numbers alone, you have to account for where each one points.

The way through is to break each force into two parts: how much of it points along the horizontal x-direction, and how much along the vertical y-direction. Those parts do add as plain numbers, because they lie along the same lines. Add up all the horizontal parts, add up all the vertical parts, and you have the resultant split into two clean components, which you can then recombine into a single force.

The equations it uses

For each force F at angle θ, the horizontal and vertical parts are F·cosθ and F·sinθ. The calculator sums those across every force to get the components of the resultant:

F_x = Σ F cosθ and F_y = Σ F sinθ

From those two components it finds the size of the net force with the Pythagorean theorem, and its direction with the arctangent:

F = √( F_x² + F_y² ) and θ = arctan( F_y ÷ F_x )

The result is the one force whose horizontal and vertical pulls match the combined pull of everything you entered.

When the net force is zero

A net force of zero is a special and important case. If all the forces on an object cancel out, the resultant is nothing, and by Newton's first law the object does not accelerate: if it was still it stays still, and if it was moving it keeps moving steadily in a straight line. This balance is called equilibrium.

It is why a book resting on a table does not fall, even though gravity pulls it down: the table pushes up with an equal and opposite force, and the two sum to zero. Whenever the components in both directions add to nothing, the object is in balance, and the calculator will return a resultant of zero.

Units and precision

Each force can be entered in newtons, kilonewtons, pounds-force, or dynes, and angles in degrees or radians, with the calculator converting to newtons and radians internally. The resultant's components and magnitude are reported in your chosen force unit and its direction as an angle from the positive x-axis. Results are carried to many figures, finer than the inputs usually justify, so the precision is in the arithmetic rather than the measurement.

A worked example: 30 N and 40 N at a right angle

Take two forces on one object: 30 N pointing along the x-axis at 0 degrees, and 40 N pointing straight up at 90 degrees.

The horizontal parts add to 30 × cos0 + 40 × cos90 = 30 + 0 = 30 N. The vertical parts add to 30 × sin0 + 40 × sin90 = 0 + 40 = 40 N. The size of the resultant is √(30² + 40²) = √2,500 = 50 N, and its direction is arctan(40/30) ≈ 53 degrees above the x-axis. The two forces of 30 and 40 combine not into 70 but into 50, pointing off at an angle.

Questions people ask

What is net force?

It is the single force equivalent to all the forces acting on an object combined. Because forces are vectors, it is found by adding them as arrows, not as plain numbers.

Why can't I just add the forces together?

Because they have directions. Forces along the same line add or subtract, but forces at an angle combine to something smaller than their plain sum. You add their horizontal and vertical parts separately.

What does a net force of zero mean?

It means the forces cancel and the object is in equilibrium. It will not accelerate: a resting object stays at rest, and a moving one continues at constant velocity.

What are the components of the net force?

They are how much of the resultant points along the horizontal x-direction and the vertical y-direction. Together they give the net force's size and direction.

References

A quick note on where the physics comes from. Treating forces as vectors, adding them by components, and the link between a zero net force and equilibrium are standard mechanics, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. The newton and the other SI units follow the US National Institute of Standards and Technology.

1. OpenStax, University Physics Volume 1, Section 5.3, Newton's Second Law of Motion (net force). https://openstax.org/books/university-physics-volume-1/pages/5-3-newtons-second-law-of-motion
2. HyperPhysics, Georgia State University, Newton's Laws and Force Vectors. http://hyperphysics.phy-astr.gsu.edu/hbase/newt.html
3. National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811