How to Calculate Net Force from a Free-Body Diagram

A free-body diagram shows the forces acting on one chosen object. To calculate the net force, add those forces as vectors: forces in opposite directions can cancel, and angled forces contribute different amounts along each axis.

The method is to choose axes, resolve any angled forces into components, and add the components with signs. The resulting vector is the net force, which determines the object’s acceleration through Newton’s second law.

What the net force means

The net force, written $\vec{F}_{\text{net}}$, is the vector sum of every external force acting on the object:

$$\vec{F}_{\text{net}}=\sum_i \vec{F}_i.$$

A free-body diagram is useful because it keeps the forces on the object separate and visible. Draw one arrow for each force, such as gravity, a normal force, tension, friction, or an applied push. The arrow’s direction shows the force direction; its length may represent the force’s size if the diagram is drawn to scale.

The net force is not necessarily in the direction the object is moving. It points in the direction of the acceleration. If the net force is zero, the object has no acceleration; it may be at rest or moving at constant velocity.

A step-by-step method

1. Isolate one object. Draw only the object being analyzed, and include the external forces acting on it. Do not include forces that the object exerts on other objects.

2. Choose axes. For a flat surface, horizontal and vertical axes are usually convenient. On an incline, axes parallel and perpendicular to the slope often simplify the calculation.

3. Resolve angled forces. For a force of magnitude $F$ at angle $\theta$ measured counterclockwise from the positive $x$-axis, its components are

   $$F_x=F\cos\theta,\qquad F_y=F\sin\theta.$$

   Assign positive or negative signs based on the direction of each component. If the angle is measured from the vertical instead, take care to use the correct component for sine and cosine.

4. Add components separately. Sum all horizontal components to get $F_{\text{net},x}$ and all vertical components to get $F_{\text{net},y}$. Opposing forces therefore subtract through their signs.

5. Combine the results. In two dimensions, calculate the magnitude and direction with

   $$|\vec{F}_{\text{net}}|=\sqrt{F_{\text{net},x}^2+F_{\text{net},y}^2},\qquad \theta=\operatorname{atan2}(F_{\text{net},y},F_{\text{net},x}).$$

   The two-argument function $\operatorname{atan2}$ gives the correct quadrant. If you use ordinary inverse tangent, check the signs of both components to choose the direction correctly.

Worked example: forces in two directions

A box on a level, frictionless surface is pushed by a $10\,\text{N}$ force east, a $6\,\text{N}$ force west, and an $8\,\text{N}$ force north. Find the net force in the plane of the surface.

Take east as positive $x$ and north as positive $y$. The horizontal and vertical components are

$$F_{\text{net},x}=10-6=4\,\text{N},\qquad F_{\text{net},y}=8\,\text{N}.$$

The net-force magnitude is

$$|\vec{F}_{\text{net}}|=\sqrt{(4\,\text{N})^2+(8\,\text{N})^2} =\sqrt{80}\,\text{N}\approx 8.94\,\text{N}.$$

Its direction, measured from east toward north, is

$$\theta=\tan^{-1}\left(\frac{8}{4}\right)\approx 63.4^\circ.$$

So the net force is about $8.94\,\text{N}$ at $63.4^\circ$ north of east. Gravity and the surface’s normal force act perpendicular to the surface and cancel if there is no vertical acceleration; they do not change this in-plane result. If the box has mass $2.0\,\text{kg}$, its acceleration in the plane is $|\vec{a}|=|\vec{F}_{\text{net}}|/m\approx 4.47\,\text{m/s}^2$, in the same direction as the net force.

Common mistakes

• Adding magnitudes without directions. A $10\,\text{N}$ force east and a $6\,\text{N}$ force west produce $4\,\text{N}$ east, not $16\,\text{N}$.
• Forgetting component signs. Decide which directions are positive and use that convention consistently. A force pointing left or down has a negative component if right and up are positive.
• Treating a diagonal force as entirely horizontal or vertical. Resolve it into both components unless it lies along an axis.
• Mixing forces on different objects. Include only forces acting on the object represented by the diagram. A force the object exerts on something else belongs on that other object’s free-body diagram.
• Confusing net force with velocity. The net force gives the direction of acceleration, not necessarily the direction of travel. An object moving right can have a net force to the left if it is slowing down.

When the net force is zero

If the signed components sum to zero on every axis, then $\vec{F}_{\text{net}}=\vec{0}$. The forces are balanced, so Newton’s second law gives $\vec{a}=\vec{0}$. This does not mean there are no forces; it means their vector sum is zero. For example, a book resting on a horizontal table has gravity downward and a normal force upward, with equal magnitudes when it has no vertical acceleration.

Once you have the net force, use Newton’s second law, $\vec{F}_{\text{net}}=m\vec{a}$, to find acceleration if the object’s mass is known. Mass is not needed to calculate the net force itself.