Net Force Formula: ΣF = ma, Adding Forces and Examples

The net force formula is $F_{\text{net}} = \sum F = ma$ : add every force acting on a body, taking direction into account, and the total equals mass times acceleration. If a rope pulls a sled forward with 70 N while snow drags it back with 20 N, the net force is 50 N forward.

Net force is the single force that would have the same effect as all the real ones together. When it is zero, the body keeps a steady velocity, which is the first law. When it is not zero, the body accelerates in its direction, which is the second law. Both ideas come from the laws of motion, and the calculator below handles the $F = ma$ step.

Quick facts

Formula
$F_{\text{net}} = \sum F = ma$
SI unit
newton (N)
Dimensional formula
[M¹ L¹ T⁻²]
Type
Vector, it sets the direction of the acceleration
Zero net force
Equilibrium: at rest or moving at constant velocity
Components
$F_x = \sum F_x$ , $F_y = \sum F_y$

What the net force formula says

Symbols in the net force formula
Symbol	Meaning	Unit
$F_{\text{net}}$	The overall, unbalanced force	N
$\sum F$	Sum of all forces, each with its sign or direction	N
$m$	Mass of the body you drew	kg
$a$	Acceleration of that body	m/s²

Some books write it $\vec{F}_{\text{net}} = m\vec{a}$ , others $\sum \vec{F} = m\vec{a}$ . They are the same law. The sum includes only forces acting on the body, never forces the body exerts on something else. That one rule settles most confusion about third law pairs.

Adding forces along one line

Pick a positive direction and give every force a sign. Then add.

Example 1: a tug of war

Team A pulls a rope right with 520 N. Team B pulls left with 480 N. Find the net force on the rope.

$F_{\text{net}} = 520 + (-480) = 40\ \text{N, to the right}$

The rope and teams accelerate slowly toward team A.

Example 2: pulling a box against friction

A 6.0 kg box is pulled across a floor with a 30 N horizontal force. Friction is 12 N. Find the acceleration.

Net force: $30 - 12 = 18\ \text{N}$ .

$a = \frac{F_{\text{net}}}{m} = \frac{18}{6.0} = 3.0\ \text{m/s}^2$

The vertical forces, weight and the normal force, cancel, so only the horizontal ones count here.

Formula for net force in a lift

Vertical problems work the same way. Take up as positive. The cable pulls up with tension T and gravity pulls down with weight mg, so $F_{\text{net}} = T - mg$ .

Example 3: lift starting upward

An 800 kg lift is pulled up by a cable tension of 9000 N. Taking g as 9.8 m/s², find the net force and the acceleration.

Weight: 800 × 9.8 = 7840 N.

$F_{\text{net}} = 9000 - 7840 = 1160\ \text{N, upward}$

$a = \frac{1160}{800} = 1.45\ \text{m/s}^2 \approx 1.5\ \text{m/s}^2$

The tension force formula page reverses this, finding the cable force from a known acceleration.

Forces at an angle: use components

When forces are not in one line, split each into x and y parts with $F\cos\theta$ and $F\sin\theta$ , where θ is measured from the x axis. Add all the x parts and all the y parts. The size of the net force is then $F_{\text{net}} = \sqrt{F_x^2 + F_y^2}$ , and its direction is $\tan\alpha = \frac{F_y}{F_x}$ .

For example, 10 N pulling east and 10 N pulling north give $F_x = 10$ , $F_y = 10$ and a net force of $\sqrt{200} = 14.1\ \text{N}$ toward the north east. For two forces with a known angle between them, the parallelogram rule on the resultant force formula page is quicker.

Zero net force and balanced forces

If $\sum F = 0$ , then a = 0, so velocity does not change. A book resting on a desk and a car cruising at constant speed on a straight road both have zero net force. In the car's case the engine's forward push through the tyres exactly matches air drag and rolling resistance.

Balanced forces do not mean no forces. They mean the forces cancel. Problems on hanging signs, ladders and stretched ropes all begin with $\sum F_x = 0$ and $\sum F_y = 0$ .

Common net force mistakes

Adding sizes without signs. 520 N and 480 N in opposite directions give 40 N, not 1000 N.
Including a reaction force. The force a box exerts on the floor does not belong in the box's sum.
Leaving out weight. In any vertical motion, gravity is one of the forces.
Assuming motion means net force. Constant velocity needs zero net force.

Draw a free body diagram every time, even when the problem looks easy. It takes thirty seconds and catches the missing force.

For the basics of a single force see the force formula page, and for friction values see the friction force formula.

Frequently asked questions

What is the formula for net force?

Net force is the vector sum of all forces on a body, F_net = ΣF. By Newton's second law it also equals ma.

How do you calculate net force with friction?

Subtract the friction force from the applied force when they act in opposite directions along the surface. Then any remaining vertical forces must be checked separately.

What happens when net force is zero?

The acceleration is zero. The body stays at rest or keeps moving in a straight line at constant speed.

Is net force the same as resultant force?

Yes, for forces on one body they mean the same thing: the single force equal to the vector sum of all the forces. Resultant is the word more often used when combining two or more given vectors.

Can net force be negative?

Along a chosen axis, yes. A negative net force simply points in the negative direction you picked.