Resultant Force

In the real world, almost nothing feels just one force at a time. Right now a football sitting on the grass is pulled down by gravity, pushed up by the ground, and squeezed on every side by the air. Kick it, and you add a huge forward shove while friction and air resistance fight back. Several forces, all at once, all on the same ball.

So how do we ever work out what the ball will do? We do something wonderfully tidy: we replace all of those forces with a single force that would have exactly the same overall effect. That one stand-in force is called the resultant force (you'll also hear it called the net force). Find the resultant, and you know everything about how the object will move — because as far as the motion is concerned, the resultant is the force.

Adding forces along a line

Forces are measured in newtons, written N. When every force acts along the same straight line — all left-and-right, or all up-and-down — finding the resultant is just careful arithmetic, and there are only two rules:

Forces pointing the same way? Add them. Two people pushing a stalled car the same way, 400 N and 300 N, give a resultant of 400 + 300 = 700 N forwards.
Forces pointing opposite ways? Subtract the smaller from the bigger. The resultant points the way the bigger force pointed.

The neat trick is to pick a positive direction first — say "right is positive" — then a force to the right counts as + and a force to the left counts as −. Now you simply add them all up, signs and all, and the sign of the answer tells you which way the resultant points.

F_{\text{resultant}} = F_1 + F_2 + F_3 + \dots

For a crate with 500 N pulling right and 200 N pulling left, that reads

(+500\ \text{N}) + (-200\ \text{N}) = +300\ \text{N},

a resultant of 300 N to the right. The same rule handled the balanced tug-of-war you met before — this is just that idea, given its proper name and a plus sign.

Force diagrams: draw every arrow first

Before you can add forces up, you have to see them. Scientists draw a free-body diagram (also called a force diagram): the object shrunk to a simple box or dot, with one labelled arrow for every force acting on it. A longer arrow means a bigger force; the arrow points the way the force pushes or pulls.

The usual cast of characters is worth knowing by name:

Weight — gravity's pull, always drawn straight down.
Normal (support) force — a surface pushing back up on whatever rests on it, at right angles to the surface.
Thrust / drive — the forward push of an engine, a rocket, or your legs.
Drag / air resistance — a fluid (air or water) pushing back against whatever moves through it, always opposing the motion.
Friction — the grip between two touching surfaces, again opposing sliding.

Once every arrow is on the page, the resultant is nothing more than adding the ups against the downs and the lefts against the rights.

Direction decides add or subtract. Two 300 N forces the same way make a 600 N resultant; the opposite way they cancel to 0 N. The numbers are identical — only the arrows differ — so always check which way each one points before you touch the arithmetic.
A resultant of zero does not mean "no forces". A book on a table has a big weight pulling down and an equally big support pushing up. The resultant is zero, but the two forces are very much still there — they simply cancel. Zero resultant means the forces are balanced, not absent.
Only draw forces on the RIGHT object. On a free-body diagram of a box, draw forces acting on the box — not the forces the box pushes back on the table or the ground. Mixing up "forces on it" with "forces it causes" is the classic slip.

Worked examples in newtons

Example 1 — a sledge, two dogs, same direction. Two husky dogs pull a sledge forwards, one with 250 N and the other with 300 N. Same direction, so add:

250\ \text{N} + 300\ \text{N} = 550\ \text{N} \ \text{forwards.}

Example 2 — a car fighting drag. A car's engine drives it forwards with 1200 N while air resistance and friction together drag back with 900 N. Opposite directions, so subtract:

1200\ \text{N} - 900\ \text{N} = 300\ \text{N} \ \text{forwards.}

The resultant is 300 N in the direction of travel — not zero, so the car speeds up.

Example 3 — a tug-of-war that isn't a tie. The right team pulls with 800 N, the left team with 950 N. Taking right as positive:

(+800\ \text{N}) + (-950\ \text{N}) = -150\ \text{N}.

The minus sign says the resultant points left: 150 N towards the left team, so the flag lurches their way. The sign did the hard work of telling us the direction.

Try it: two pushes, one resultant

Here is a box with two forces acting along one line: a push to the right and a push to the left, each set in newtons by a slider. The purple resultant arrow below the box shows what the two together add up to — its length is the difference between the pushes, and it points towards the stronger side. Make the two pushes equal and the resultant vanishes: the box is balanced. Try 500 N against 200 N and read off the 300 N leftover.

Forces at right angles (a peek ahead)

What if two forces don't lie along the same line, but pull at right angles — say one north and one east? You can't just add or subtract them, because they aren't fighting or helping along a single line. Instead you draw them as two sides of a right-angled triangle, and the resultant is the slanting third side — the hypotenuse — found with Pythagoras.

A boat is driven 4000 N forwards by its engine while a river current shoves it 3000 N sideways. The resultant is

F = \sqrt{4000^2 + 3000^2} = \sqrt{25\,000\,000} = 5000\ \text{N},

pointing off at an angle between the two — a tidy 3-4-5 triangle. You'll meet this properly when you study vectors; for now, just remember that same-line forces add or subtract, but forces at an angle build a triangle.

Watch a little tug boat nudge a container ship a thousand times its size into harbour. How can something so small move something so vast? It can't — not on its own. The trick is that the ship is already floating almost perfectly balanced, its huge weight exactly matched by the water's upthrust, so along the surface the resultant force is nearly zero. The tug only has to supply a tiny extra resultant, and with almost nothing to cancel it, that little push is enough to swing the giant round — slowly, patiently, over many minutes. It is a perfect lesson in resultant force: it isn't the size of a single push that matters, but what's left over once all the forces are added up.