Net force: what it is, the formula (add the forces as vectors, then F = ma), balanced against unbalanced forces, how to draw a free-body diagram, how to find the net force in one and two dimensions, ten worked examples, the mistakes that lose marks, and a printable sheet with practice

"Net force" is searched about 30,000 times a month in the United States alone, mostly by middle-school students (NGSS puts it in Grades 6 to 8) and their parents, and the phrase "balanced and unbalanced forces" another 4,000. The idea is small: add up every force on an object, direction included, and the total tells you whether the object's motion changes. The marks are lost in the details: a minus sign, a force drawn on the wrong object, the belief that a moving object must have a net force on it. This page sets out the formula, the balanced-against-unbalanced distinction, the four-step free-body diagram, the one- and two-dimensional cases, ten worked examples and the eight common mistakes, with a printable sheet of diagrams and practice.

Download the net force sheet with practice (PDF) (A4, the rules, the diagrams and the worked examples on page 1, the practice set with answers on page 2, free).

What net force is

Every object has several forces acting on it at once: its weight pulling down, a surface pushing up, a hand pushing, friction dragging, air resisting. The net force (the resultant force in British textbooks) is the single force that would have exactly the same effect as all of them together. It is a vector: it has a size in newtons and a direction.

The one sentence to remember, from Newton's first and second laws:

If the net force is zero, the motion does not change. If the net force is not zero, the object accelerates in the direction of the net force, by a = F net ÷ m.

"Motion does not change" covers two cases that children treat as different: staying still, and moving at a constant speed in a straight line. Both are zero net force.

The formula

In a straight line. Choose a positive direction (right, or up). Forces in that direction are positive, forces the other way negative. Add them.

F net = F1 + F2 + F3 + ... (with signs)

• A 10 N push to the right and 4 N of friction to the left: F net = 10 − 4 = 6 N to the right.
• Two children pulling a rope with 50 N each, opposite ways: F net = 50 − 50 = 0 N. Balanced.
• A 20 N push right and a 25 N push left: F net = 20 − 25 = −5 N, that is 5 N to the left.

Then Newton's second law. F net = m × a, so

a = F net ÷ m

A 6 N net force on a 2 kg box gives a = 6 ÷ 2 = 3 m/s². The same 6 N on a 12 kg box gives 0.5 m/s². The acceleration is always in the direction of the net force, which is not always the direction the object is moving (a ball thrown upward is moving up while the net force, its weight, is down; that is why it slows).

At right angles. Two perpendicular forces combine by Pythagoras: F net = √(Fx² + Fy²), with the direction from tan θ = Fy ÷ Fx. A 3 N push east and a 4 N push north give √(9 + 16) = 5 N, at 53° north of east.

At any angle. Split each force into horizontal and vertical parts (F cos θ and F sin θ), add the horizontals, add the verticals, then Pythagoras. This is the Grade 11 and A-level version; Grades 6 to 9 stay in a line or at right angles.

Balanced against unbalanced forces

The test question that separates children who understand from children who memorised: a car travels along a straight motorway at a constant 100 km/h. Are the forces on it balanced? Yes. The engine's forward force equals friction plus air resistance; the net force is zero; the velocity does not change. "It is moving, so there must be a net force" is the mistake, and Newton's first law exists to correct it.

How to draw a free-body diagram

1. Draw the object as a dot or a box. Nothing else: no table, no floor, no rope, no other object.
2. Draw one arrow for each force acting ON the object, starting at the dot and pointing the way the force acts. Longer arrow, bigger force.
3. Label each arrow with the force's name and, if known, its size: weight (W or mg), normal force (N, the surface's push), friction (f), tension (T), applied force (F), air resistance or drag.
4. Check: every arrow must be a force something else exerts on this object. Forces the object exerts on other things (its push on the floor) do not go on its diagram. "Velocity" and "motion" are not forces and never get an arrow.

Then add the arrows: horizontals together, verticals together.

The forces that appear

Ten worked examples

1. Book on a table. Weight 15 N down, normal force 15 N up. F net = 0. Balanced; the book stays put.
2. Dropped ball (no air resistance). Weight 2 N down, nothing up. F net = 2 N down. Unbalanced; a = 2 ÷ 0.2 = 10 m/s² (the acceleration of free fall).
3. Box pushed on a rough floor. Push 40 N right, friction 15 N left, weight and normal force cancel vertically. F net = 25 N right. For a 5 kg box, a = 5 m/s².
4. Tug of war. Left team 800 N, right team 950 N. F net = 150 N toward the right team. The rope (and the losing team) accelerate that way.
5. Lift (elevator) moving up at a steady speed. Weight 600 N down, cable tension 600 N up. F net = 0: steady speed means balanced, even though it is moving.
6. Lift accelerating upward at 2 m/s². Mass 60 kg, weight 588 N. F net = ma = 120 N up, so the tension (or the floor's push) is 588 + 120 = 708 N. The person feels heavier.
7. Car cruising at 90 km/h. Engine force 2,000 N forward, air resistance 1,400 N and rolling friction 600 N backward. F net = 0. Constant speed.
8. Car braking. Braking force 6,000 N backward, no engine force. F net = 6,000 N backward; for a 1,200 kg car, a = −5 m/s² (slowing).
9. Skydiver at terminal velocity. Weight 750 N down, air resistance 750 N up. F net = 0; falls at a constant speed. Before terminal velocity, air resistance is smaller than the weight, the net force is down, and the skydiver is still speeding up.
10. Sled pulled by two ropes at right angles. 30 N north and 40 N east. F net = √(30² + 40²) = 50 N, at 37° north of east (tan θ = 30 ÷ 40). On a 25 kg sled with no friction, a = 2 m/s² in that direction.

The eight mistakes that lose marks

1. Thinking a moving object must have a net force on it (constant velocity means zero net force).
2. Adding forces without signs: 10 N and 4 N in opposite directions is 6 N, not 14 N.
3. Drawing the force the object exerts (its push on the ground) on the object's own diagram.
4. Giving "motion" or "speed" an arrow.
5. Making the normal force always equal to the weight. It is only equal on a flat surface with no other vertical forces; on a slope or with a hand pressing down it is different.
6. Forgetting that friction opposes the motion, so it is negative when the object moves in the positive direction.
7. Confusing mass (kg) and weight (N). F net = ma needs the mass; weight = mg.
8. Adding perpendicular forces as if they were in a line (3 N and 4 N at right angles make 5 N, not 7 N).

Where it sits in each curriculum

Practice (answers in the PDF)

1. A 12 N push to the right and a 7 N friction force to the left act on a box. What is the net force? Is the box's motion changing?
2. A 3 kg object has a net force of 18 N on it. What is its acceleration?
3. A parachutist falls at a constant 5 m/s. Her weight is 700 N. What is the air resistance on her, and what is the net force?
4. Two forces of 6 N and 8 N act on an object at right angles. What is the net force?
5. A 1,000 kg car accelerates at 3 m/s². Air and friction forces total 500 N. What forward force must the engine supply?
6. Draw the free-body diagram for a book sliding to the right across a table and slowing down. Which way does the net force point?

Download the sheet and the practice set (PDF)

Where to get help

A 1-on-1 online physics class with a tutor who teaches NGSS middle-school science, KS3 and GCSE physics, CBSE Class 8 and 9 and the Australian Years 7 to 10 every week: forces taught by drawing, every problem starting with a free-body diagram and a chosen positive direction, so the signs look after themselves. The physics page lists the courses; the free 30-minute first class is a real lesson on this week's homework. The next pages in this set are Newton's laws of motion, the five equations of motion and types of motion and acceleration.

Questions parents ask

See how we teach this, 1-on-1 online →