Moments
Try to push a heavy door open by pressing right next to its hinges. It barely moves, however
hard you shove. Now press on the far edge, where the handle is — the same push swings the whole
door open with ease. The force is identical. What changed is where you pushed.
That is the whole idea of a moment: the turning effect of a
force about a fixed point called the pivot (or fulcrum). Whenever a force acts
at a distance from a pivot, it doesn't just push — it turns. A spanner turns a bolt, a
see-saw tips, a wrench cracks open a jar lid, your forearm swings about your elbow. Every one of
these is a moment at work.
Two things decide how big that turning effect is: how hard you push (the force),
and how far from the pivot you push (the distance). Make either one bigger and
the turn gets stronger. That single sentence is what this whole page unpacks.
Measuring the turn: M = F\,d
We can put a number on the turning effect. The moment of a force is simply the
force multiplied by its distance from the pivot:
M = F \, d
Here F is the force in newtons (N),
d is the perpendicular distance from the pivot in
metres (m), and the moment M comes out in
newton-metres (N·m) — a force unit times a distance unit. Read it aloud as
"force times distance."
Because it is a product, you get a bigger moment two ways — and they count equally:
- A bigger force. Push twice as hard at the same spot and you double the turn.
- A longer distance. Push at twice the distance with the same force and you
also double the turn.
This is exactly why a door handle is fitted at the edge farthest from the hinge, why a
long spanner loosens a rusted bolt that a short one can't budge, and why you
instinctively grab the end of a broom to sweep. A little more distance is free extra turning
power.
Worked example. You push a door handle with a force of
F = 5\text{ N}, and the handle is d = 0.8\text{ m}
from the hinges. The moment is
M = F\,d = 5 \times 0.8 = 4 \text{ N·m}.
Now imagine pressing the same 5 N right beside the hinges, only
0.05\text{ m} out:
M = 5 \times 0.05 = 0.25\text{ N·m} — sixteen times feebler. Same
force, tiny distance, tiny turn. That is the door refusing to budge.
Have a go: balance the see-saw
On the left a fixed weight makes an anticlockwise moment of
40 \times 2 = 80\text{ N·m}. On the right, you choose the
force and how far out it sits, making a clockwise moment. Watch the readout: when
your clockwise moment is smaller the beam tips left, when it is larger it tips right, and when the
two moments are exactly equal the beam sits level and balanced. Notice there is
more than one way to balance it — a big force close in, or a small force far out.
The Principle of Moments
When an object is balanced — not turning, in what we call
equilibrium — there must be just as much turn one way as the other. Anything else
and it would start to spin. This gives us a beautifully simple rule for anything that balances on
a pivot.
For an object balanced (in equilibrium) about a pivot:
- the total clockwise moment equals the
total anticlockwise moment about that pivot;
- in symbols, F_1 d_1 = F_2 d_2 (summed over every force on each
side), with each distance measured perpendicular from the pivot.
This one equation lets you find a missing force or a missing distance. If three of the four
quantities are known, the fourth follows.
Worked example — a missing distance. A child weighing
300\text{ N} sits 1.5\text{ m} to the left
of the pivot. An adult weighing 450\text{ N} sits on the right. Where
must the adult sit to balance the see-saw?
Set the clockwise moment equal to the anticlockwise moment:
450 \times d = 300 \times 1.5 = 450 \text{ N·m} \;\;\Rightarrow\;\; d = \frac{450}{450} = 1 \text{ m}.
The heavier adult must sit closer to the pivot — only 1 m out — to match the lighter
child sitting 1.5 m out. Heavier means nearer the pivot; lighter means farther away. That is why
two children of different sizes shuffle along a see-saw until it balances.
-
Measure the distance from the pivot, and make it perpendicular. The
d in M = F\,d is the perpendicular
distance between the pivot and the line of the force — not the length of the arm, and
not the distance to some other point. Measure to the wrong point and every moment is wrong.
-
A force right at the pivot has no turning effect at all. If
d = 0 then M = F \times 0 = 0, no matter
how huge the force. This is why nobody sits in the middle of a see-saw, and why pushing a door
on its hinges does nothing.
-
A moment always needs BOTH a force and a distance. A force on its own is just
a push; a distance on its own does nothing. Only a force acting at a distance makes a
moment.
Levers: turning a small force into a big one
The Principle of Moments is the secret behind every lever — a stiff bar that
turns on a pivot. Because M = F\,d, a small force acting at a
large distance can balance (and lift) a large force acting at a small distance.
Put your effort far from the pivot and the load close to it, and the lever multiplies your
strength.
Worked example — a crowbar. A crowbar rests on a pivot. A heavy crate presses
down with 800\text{ N} at just 0.1\text{ m}
from the pivot. You push down on the far end, 0.8\text{ m} from the
pivot. What force do you need to lift it?
F \times 0.8 = 800 \times 0.1 = 80 \text{ N·m} \;\;\Rightarrow\;\; F = \frac{80}{0.8} = 100 \text{ N}.
Just 100\text{ N} from you lifts an 800\text{ N}
crate — the lever has multiplied your force eight times, because your distance
from the pivot is eight times the load's. A wheelbarrow (wheel = pivot, load near it, you lift at
the far handles) and a pair of scissors work in exactly the same way.
There is no magic and nothing for free, though: to lift the crate a little, your end of the bar
must sweep through a much longer arc. You trade a big, easy movement of your hand for a
small, powerful movement of the load — force multiplied, distance divided.
High above the crowd, a tightrope walker holds a pole several metres long — and it is doing
serious physics. The moment you start to topple, gravity pulls you round the rope with a small
turning moment. To fight it you need an opposing moment, fast. A long pole
gives you enormous d: by nudging its far ends only slightly the
walker creates a large restoring moment about the rope, gently turning their body upright again
before the fall runs away with them.
The pole is often weighted at the tips and it droops downward, which lowers the walker's centre
of mass too — but the heart of the trick is pure moments: a little force, applied a long way
out, buys a lot of turning control. Take the pole away and the same wobble becomes a fall.
The whole idea in one breath
- A moment is the turning effect of a force about a pivot:
M = F\,d, measured in N·m.
- Bigger force or longer (perpendicular) distance → bigger moment.
- When something is balanced, clockwise moments = anticlockwise moments — the
Principle of Moments.
- Levers use this to turn a small force at a large distance into a large force at a small
distance.