Free-Body Diagrams & Net Force · Grade 12 Science

Free-Body Diagrams & Net Force

Learning goals

• I can draw a free-body diagram that shows every force acting on a single object as a labeled vector.
• I can compute the net force from several forces, including forces at right angles, and use ΣF = ma to find the resulting acceleration.
• I can recognize when an object is in equilibrium because its forces sum to zero.

Readings for this unit

• The force on the box is not the force on the floor

Real objects rarely feel just one force. A crate being dragged across a warehouse floor feels its own weight pulling down, the floor pushing up, a rope pulling forward, and friction dragging back — four forces at once. To predict how it moves, you cannot juggle all of that in your head; you need a system. That system is the free-body diagram, the single most-used tool in all of mechanics, and it is the bridge between a messy real scene and the clean equation ΣF = ma.

To draw one, you mentally cut the object free from its surroundings and represent it as a simple dot or box. Then you draw an arrow for every force *acting on that object* — and only those. Each arrow starts at the object, points the way the force pushes or pulls, and is drawn longer for a stronger force. The usual cast of characters is small and worth memorizing: weight (gravity, always straight down), the normal force (a surface pushing perpendicular to itself), tension (a rope or cable pulling along its length), an applied force (a direct push or pull), and friction (always opposing sliding). Naming the forces correctly is more than half the battle.

Once every force is on the diagram, you find the net force by adding the arrows as vectors — quantities with both size and direction. Forces along the same line are easy: add them if they point the same way, subtract them if they oppose. The net force is what actually goes into Newton's second law, which in its full form reads ΣF = ma. A common and costly mistake is to plug a single force into that equation; only the *sum* of all forces determines the acceleration, and the object always accelerates in the direction of that net force.

Because forces are vectors, it usually pays to split the problem into directions and handle each independently. Physicists resolve forces into perpendicular components, typically horizontal (x) and vertical (y), then apply Newton's second law separately to each: ΣFₓ = maₓ and ΣF_y = ma_y. For a crate sliding along level ground, the vertical forces (weight down, normal up) cancel, so all the acceleration comes from the horizontal forces. Breaking a two-dimensional problem into two one-dimensional problems is the standard professional move, and it keeps the arithmetic honest.

When two forces act at a right angle instead of along one line, you combine them with the Pythagorean theorem, exactly as in geometry. A 9 N pull east and a 12 N pull north do not add to 21 N; the true net force is √(9² + 12²) = √225 = 15 N, aimed diagonally between them. This is a beautiful moment where math and physics turn out to be the same tool: the resultant of two perpendicular forces is literally the hypotenuse of a right triangle whose legs are those forces.

A special and important case is equilibrium, when all the forces sum to exactly zero. A book resting on a table, a lamp hanging motionless from a cord, and an airplane cruising at a steady altitude and speed are all in equilibrium — not because no forces act, but because the forces perfectly cancel. By the first law, zero net force means the velocity never changes, so an object in equilibrium is either sitting still or moving at constant velocity. Reading a free-body diagram tells you instantly which case you are in: if the arrows balance, motion holds steady; if they don't, the leftover net force sets the acceleration through ΣF = ma.

Model check: Draw only forces exerted on the selected object; motion arrows are not forces. Net 12N right; a=12/4=3 m/s² right.

Net Force (same line)

ΣF = F₁ + F₂ + …

add forces pointing the same way; subtract forces pointing opposite ways, to get the net force along one direction.

Resultant of Perpendicular Forces

F net = √(Fₓ² + F_y²)

Fₓ and F_y are two forces (or force components) at right angles; their vector sum has this magnitude.

Worked examples

Avoid these mistakes

• Draw only forces exerted on the selected object; motion arrows are not forces.

Words to know

free-body diagram

A simplified sketch of a single object with an arrow drawn for every force acting on it, pointed in the correct direction and sized to show relative strength.

net force (ΣF)

The single overall force found by adding every force on an object as a vector, accounting for direction.

equilibrium

The condition in which the net force on an object is zero, so its velocity does not change.

normal force

The support force a surface pushes back with, always directed perpendicular to the surface.

vector

A quantity that has both magnitude and direction, such as force, velocity, or acceleration.

The free-body diagram: mechanics' most-used tool

Ask any physicist or engineer for the first step in a force problem and they will say the same thing: draw a free-body diagram. The technique is powerful precisely because it forces you to make three decisions explicit — which object you are analyzing, which forces act on it (and which do not), and which direction each one points. A frequent beginner mistake is to include forces the object exerts on other things; a free-body diagram shows only the forces acting on the chosen object itself. Done carefully, it converts a confusing real-world scene into a clean set of vectors ready for ΣF = ma.

Resolving forces into components

Because forces are vectors, the professional approach to any two-dimensional problem is to choose perpendicular axes (often x and y, sometimes tilted to match a ramp) and break every force into components along them. Then Newton's second law is applied separately in each direction: ΣFₓ = maₓ and ΣF_y = ma_y. A force F at angle θ splits into F cosθ and F sinθ, the same trigonometry used throughout geometry. This is why a rope pulled at an angle is less effective at dragging a sled than a horizontal pull — part of its force goes into lifting rather than pulling, and only the horizontal component contributes to forward motion.

Constant velocity is not the same as 'no forces'

It is tempting to think that if an object's engine is running, it must be accelerating — but an airplane cruising at a steady altitude and speed is in perfect equilibrium. Lift balances weight, and the engines' thrust balances air drag; the net force is zero even though four large forces are acting. The engines run constantly not to speed the plane up, but simply to cancel the drag trying to slow it down. The same logic explains terminal velocity: a skydiver stops accelerating once air resistance grows to equal their weight, after which they fall at a constant speed with zero net force.

Static equilibrium: why bridges and buildings stand

Structural engineering is, at heart, the art of keeping enormous objects in static equilibrium — every net force (and every net rotational effect, called torque) balanced to zero. Engineers draw free-body diagrams of individual joints, beams, and cables to verify that the forces of traffic, wind, snow, and a structure's own weight are exactly opposed by the supports and tension members holding it up. When a designer gets a single force balance wrong, the consequence is not a lost point on a test but a structure that can sag, crack, or collapse — which is why the humble free-body diagram is a genuine matter of public safety.

In the real world

Structural engineers draw free-body diagrams of every joint in a bridge to confirm the forces of traffic, wind, and the bridge's own weight all cancel — an unbalanced joint is a bridge that could fail.

Exit check

A4 kg block has 20N right and 8N left. Find acceleration.