Balanced Forces and Resultants

Newton's first law said that motion only changes when a net force acts, and Newton's second law needs that net force as its input. This lesson is about actually finding it — adding several forces together into the single force that would have the same effect, and recognising the important special case where they add up to nothing at all.

Forces along the same line

When forces act along one line, adding them is arithmetic with signs. Two forces pushing the same way add: a 300 newton thrust plus a 200 newton tailwind push gives 500 newtons forward. Forces opposing each other subtract: a 300 newton thrust against 120 newtons of drag leaves a net 180 newtons forward. The convention that makes this reliable is to pick one direction as positive before you start, write every force with its sign, and then simply add — that way you never have to decide whether a particular case is an addition or a subtraction, because the signs decide for you.

Equilibrium

When the forces on an object cancel exactly, the net force is zero and the object is in equilibrium. The trap in the word is that equilibrium does not mean stationary. A parachutist at terminal velocity is in equilibrium and falling at speed; an aircraft in level flight at constant speed is in equilibrium with four substantial forces acting on it. Equilibrium means unchanging motion, which covers both being at rest and moving steadily in a straight line. That is Newton's first law restated, and it is the reason the two laws are best learned together.

This gives you a powerful shortcut. Whenever a question says an object moves at constant velocity, or hovers, or slides at a steady speed, it has told you that the net force is zero — and therefore that whichever force you were asked about must be equal and opposite to the ones you know. You can often answer without touching F = ma at all.

Forces at right angles

Forces that are not along the same line cannot be added as numbers, because their directions matter. Two forces at right angles combine into a resultant found by drawing them head to tail and closing the triangle — and if you choose the numbers well, the arithmetic stays clean. A 3 and a 4 at right angles give exactly 5, because 3, 4 and 5 form a right-angled triangle. Those are the numbers questions use, precisely so that no calculator is needed.

The rule behind those numbers is Pythagoras' theorem: in a right-angled triangle, the square on the longest side equals the squares on the other two sides added together. Written out for two perpendicular forces, it gives the size of the resultant directly.

Notice what the figure rules out. The two forces total 7 newtons if you simply add the numbers, and 7 will be offered to you as an answer. It is wrong because force is a vector: pulling east and pulling north do not help each other, so the combined effect is less than the sum. The resultant also has a direction — somewhere between the two contributing forces, nearer the larger one — and a complete answer says so even when the question only asks for the size.

Putting it to work

Naming the forces in a balanced situation

Equilibrium problems are usually easier than they look, because the balance condition hands you the answer. A book resting on a table has its weight pulling down and the table's normal force pushing up, and since it is not moving these must be equal — so if you are told the weight is 8 newtons you already know the normal force is 8 newtons, without any calculation. Hang the same book from a string and the string's tension takes over the same role: tension up, weight down, equal in size. In each case one force is known and the balance condition supplies the other.

The same reasoning extends to more forces. Something sliding down a slope at steady speed has its weight, the surface's normal force and friction all acting, and because the speed is steady the whole set must cancel. Questions built this way look intimidating because of the number of arrows, but the sentence "at constant speed" is doing the work: it tells you the total is zero, so whichever arrow you were asked about is fixed by the others. Look for that phrase before starting any arithmetic.

Keeping the signs straight

When several forces act along one line, the reliable method is bookkeeping rather than judgement. Choose a positive direction and write it down. Give every force a sign according to that choice. Add them all. If the total comes out positive the resultant points the way you chose; negative and it points the other way; zero and you have equilibrium. The reason to be this mechanical about it is that questions deliberately mix the order the forces are presented in, and deciding case by case whether to add or subtract is where errors creep in. With signs, there is only ever one operation.

One last case is worth working through because it recurs: an object held still by three forces rather than two. A picture hanging from two wires has its weight pulling straight down and a tension in each wire pulling up and outwards. The two tensions must between them cancel the weight vertically, and cancel each other horizontally, which is why the wires pull in opposite sideways directions. Notice what that means for the tension: the more shallowly the wires are angled, the more of each tension is spent pulling sideways against the other wire rather than upwards against the weight, so the tension needed grows. Stringing a washing line very taut takes far more force than letting it sag, and the reason is that at a shallow angle almost none of the tension is doing the lifting.

Why it matters in aviation

The vector addition in this lesson is performed on every flight under the name of the wind triangle. The aircraft has a velocity relative to the air, given by its heading and airspeed, and the air has a velocity relative to the ground, given by the wind's direction and speed. Add the two and the resultant is the track and groundspeed actually achieved — where the aircraft goes, and how quickly. Because that is a vector sum and not an arithmetic one, a wind from the side carries the aircraft off the line it is pointed along, and the fix is to point slightly into the wind so that the two sideways components cancel and the resultant lies along the intended track.

Resolving does the same work in reverse. A reported wind is one vector, and splitting it into a component along the runway and a component across it separates two quantities a crew needs for different reasons. The headwind component reduces the groundspeed at which the aircraft leaves or touches the ground, and so shortens the distance required, while the crosswind component is compared against a published limit that decides whether that runway can be used at all. A strong wind almost down the runway and a moderate wind across it can resolve to very different answers.

What the test asks

The single most productive distractor in this topic is the arithmetic sum of two perpendicular forces — 70 for the boat above, 7 for the triangle in the figure. The second is the belief that equilibrium means stationary, which makes a steadily falling parachutist look like a contradiction. The third is subtler: being given several forces and using one of them in F = ma instead of the net force, which gives an acceleration that is too large.