Moments, Levers and Centre of Gravity - Physics for Pilot Assessments

Forces do not only push things along; they also turn them. A spanner on a bolt, a seesaw, a door on its hinges and a loaded aircraft all involve a force acting at a distance from a pivot, and the quantity that describes the turning is the moment. It is straightforward arithmetic with one detail that decides most questions: which distance you are allowed to use.

The moment of a force

A moment is a force multiplied by a distance from the pivot — but specifically the perpendicular distance, measured at right angles from the pivot to the line the force acts along.

Moment of a force

M = F d_{\perp}

$M$: the moment, in newton metres (N m)
$F$: the force, in newtons (N)
$d_{\perp}$: the PERPENDICULAR distance from the pivot to the force's line of action, in metres (m)

The unit is the newton metre, and it is worth noticing that this is a newton multiplied by a metre rather than divided by one — a moment is not a pressure and not an energy, even though the same two units appear in those quantities too. The word torque means the same thing and you will meet both.

Why perpendicular? Because only the part of the force that acts across the lever arm does any turning. Pull along the spanner, straight towards or away from the bolt, and it will not turn at all no matter how hard you pull — the perpendicular distance in that case is zero. Pull at right angles to the spanner and every bit of your force is turning it. At any angle in between, the effective distance is shorter than the spanner's full length.

Top: the force acts square-on, so the whole length of the spanner counts. Bottom: the same force at an angle, and the perpendicular distance from the pivot to its line of action is shorter — so the moment is smaller.

The principle of moments

A seesaw balances when the turning effects cancel. Anticlockwise moments equal clockwise moments, and because a moment is a force times a distance, a small force far out can balance a large force close in. That is the whole idea of a lever: it lets a modest force overcome a large one by giving it a longer arm.

Centre of gravity, and toppling

Every object behaves, for the purpose of weight, as though its whole weight acted at a single point called the centre of gravity. For a uniform ruler that point is the middle; for an oddly shaped or unevenly loaded object it can be anywhere, including outside the material itself in the case of a horseshoe. Knowing where it is matters because it decides stability.

Tilt an object and its weight still acts straight down from the centre of gravity. While that downward line falls inside the base, the weight produces a moment that rotates the object back onto its base — it is stable. Push it far enough that the line falls outside the pivoting edge, and the same weight now produces a moment the other way, and it topples. This is why a low, wide object is hard to tip and a tall, narrow one is easy, and why loading heavy items low down makes anything more stable.

Left: the weight line still falls inside the base, so the block rocks back. Right: tilted further, the line has passed the pivot edge and the same weight now turns it over.

Putting it to work

Example 3 — finding the balance point

A beam of negligible weight carries a 100 N load 1 m from its left end and a 300 N load 3 m from its left end. Where must a single support be placed for the beam to balance?

The two loads together behave as one weight of 400 N acting at their combined centre of gravity, and the support must sit exactly there.
Take moments about the left end: $100 \times 1 + 300 \times 3 = 100 + 900 = 1{,}000\ \text{N m}$
Divide by the total weight: $\frac{1{,}000}{400} = 2.5$ m from the left end.
Check it balances. From 2.5 m, the 100 N load is 1.5 m away, giving 150 N m one way; the 300 N load is 0.5 m away, giving 150 N m the other. Equal, so it balances — the heavier load gets the shorter arm, exactly as the seesaw rule demands.

Where the pivot sits

Levers come in arrangements that look quite different but obey the same rule. A seesaw or a pair of scissors has the pivot between the effort and the load. A wheelbarrow has the pivot at one end, with the load in the middle and your effort at the far end — which is why a longer-handled barrow is easier. A pair of tweezers puts the effort between the pivot and the load, which multiplies the distance moved rather than the force, and is exactly what you want for delicate work. When you meet an unfamiliar tool, find the pivot first and the rest follows.

The practical consequence of the moment relationship is that a longer arm is worth more than a stronger arm. Doubling the length of a spanner doubles the moment for the same effort, which is why a length of pipe slipped over a spanner handle frees a seized bolt that no amount of grip could shift. The same logic explains why a door handle is fitted at the edge furthest from the hinge rather than in the middle: the designer is buying the largest possible perpendicular distance.

Raising the centre of gravity

Because stability depends on the weight line staying inside the base, two things make an object easier to tip: a narrower base and a higher centre of gravity. Both appear in questions. A double-decker bus with an empty lower deck and a full upper one has moved its centre of gravity up and become measurably easier to tilt, without any change in total weight. A racing car sits as low as it can for the opposite reason. And loading a container heavy-items-first at the bottom is not merely convenient — it lowers the centre of gravity and makes the whole thing harder to overturn.

One more practical point about balance. When a question involves a beam or plank with weight of its own, that weight acts at its centre of gravity — the middle, for a uniform beam — and produces a moment like any other force. Forget it and the arithmetic will not balance. A uniform plank pivoted at one end has its own weight acting halfway along, so a plank weighing 40 newtons and four metres long contributes a moment of 40 multiplied by 2, which is 80 newton metres, before any load is placed on it. Questions that describe the beam as "light" or "of negligible weight" are telling you explicitly to skip that term; if they do not say so, it counts.

Why it matters in aviation

Everything an aircraft carries is accounted for as a moment before it flies. Each item of load — fuel in each tank, baggage in each hold, passengers by seating row — has a known arm, its distance from a fixed reference point called the datum, and weight multiplied by arm gives its moment about that datum. Add all the moments, divide by the total weight, and the result locates the centre of gravity, which must fall inside a range published for the type. It is the balance arithmetic of this lesson with more terms, and it is why loading is a calculation rather than a matter of filling holds evenly.

The limits exist because the position of the centre of gravity decides how the aircraft handles. Too far aft and it becomes less stable in pitch, more willing to diverge from the attitude it was trimmed for and slower to settle back; too far forward and the tail surface may not have the leverage to raise the nose at takeoff, or to hold it up as the aircraft slows on approach. Neither problem is about total weight, which can be comfortably within limits in both cases. It is about where that weight sits relative to the point the aircraft balances about.

What the test asks

The reliable trap is an angled force with the full length of the lever offered as the distance. If the force is not at right angles, the perpendicular distance is shorter and the moment is smaller than the naive product suggests. The second is a balance question where the two distances are measured from different points rather than both from the pivot. The third concerns stability: expect to be asked which of several shapes topples most easily, where the answer follows from base width and how high the centre of gravity sits, not from weight.