Moments and the Principle of Moments – Force and Equilibrium of Objects (Senior 1 Physics) | Ganzaa

Force and Equilibrium of Objects

Lessons

Senior 1 · Physics

Force and Equilibrium of Objects

5 lessons

Lesson 4 of 5

Moments and the Principle of Moments

25 min

Why is the handle far from the hinges?

Try to open a classroom door by pushing right next to its hinges: it is hard work. Push at the handle, far from the hinges, and the same door swings open easily. On a see-saw, a small child can balance a much heavier one by sitting further from the middle. In both cases a force is turning something about a fixed point, and the turning effect depends on how far from that point the force acts.

Definition

A pivot (or fulcrum) is the fixed point about which an object turns.

The moment of a force (its turning effect, also called torque, $\tau$ ) is the force multiplied by the perpendicular distance from the pivot to the line of the force:

\tau = F \times d

SI unit: the newton metre (N m). A moment turns either clockwise or anticlockwise.

Figure 4.1: A spanner. The force acts at right angles to the handle, 0.25 m from the pivot, so its moment is 10 N m.

Worked example 1: loosening a nut

A mechanic pushes down with 40 N on the end of a spanner, 0.25 m from the centre of the nut, at right angles to the handle (Figure 4.1). Find the moment.

\tau = F \times d = 40\,\mathrm{N} \times 0.25\,\mathrm{m} = 10\,\mathrm{N\,m} \text{ (clockwise)}

With a spanner twice as long, the same 40 N would give 20 N m. That is why a stiff nut is loosened with a long spanner.

Experiment: balancing a metre rule

Figure 4.2: A metre rule balanced on a knife edge at its centre, with weights hung on each side.

Balance a metre rule on a knife edge at its 50 cm mark. Because the rule is uniform, its own weight acts at the pivot and has no turning effect.
Hang a 2.0 N weight on a loop of thread 30 cm to the left of the pivot.
Hang a 3.0 N weight on the right and slide it until the rule balances level. Measure its distance from the pivot.
Repeat with other weights and distances, and work out each moment.

Left force (N)	Left distance (m)	Anticlockwise moment (N m)	Right force (N)	Right distance (m)	Clockwise moment (N m)
2.0	0.30	0.60	3.0	0.20	0.60
1.0	0.40	0.40	2.0	0.20	0.40
1.5	0.40	0.60	2.0	0.29	0.58

Whenever the rule balances, the anticlockwise moment equals the clockwise moment, within the error of measuring the distances.

Important

Safety. Keep the weights small (a few newtons), tie the threads firmly, and keep your feet from under the hanging weights.

Definition

The principle of moments: when an object is balanced (in equilibrium), the sum of the clockwise moments about any point equals the sum of the anticlockwise moments about the same point.

\text{sum of clockwise moments} = \text{sum of anticlockwise moments}

An object is in equilibrium when (1) the resultant force on it is zero, AND (2) the total moment about any point is zero. Then it neither starts to move nor starts to turn.

The see-saw

Figure 4.3: A balanced see-saw. The lighter child sits further from the pivot, so the two moments are equal.

Worked example 2: where should he sit?

Keza (weight 400 N) sits 1.5 m from the pivot of a see-saw. Where must Mugisha (weight 600 N) sit on the other side to balance her?

Principle of moments about the pivot:

600\,\mathrm{N} \times d = 400\,\mathrm{N} \times 1.5\,\mathrm{m} \quad\Rightarrow\quad d = \frac{600\,\mathrm{N\,m}}{600\,\mathrm{N}} = 1.0\,\mathrm{m}

Worked example 3: the wheelbarrow

A wheelbarrow carries a load of 600 N whose weight acts 0.40 m from the axle of the wheel (the pivot). The builder lifts the handles 1.2 m from the axle. What upward force must he apply?

Taking moments about the axle, the load turns the barrow one way and the lift turns it the other:

E \times 1.2\,\mathrm{m} = 600\,\mathrm{N} \times 0.40\,\mathrm{m} \quad\Rightarrow\quad E = \frac{240\,\mathrm{N\,m}}{1.2\,\mathrm{m}} = 200\,\mathrm{N}

He lifts only a third of the load, because his hands are three times as far from the pivot.

Warning

Common mistakes. (1) Using the distance along a slanting bar instead of the PERPENDICULAR distance from the pivot to the line of the force. (2) Mixing centimetres and metres: change to metres to get N m. (3) Forgetting the weight of the beam itself when it is not pivoted at its centre: a uniform beam's weight acts at its middle.

Tip

Take moments about the pivot: the pivot's own force has zero distance, so it has no moment and drops out of your equation. Then write "clockwise = anticlockwise" and solve.

Levers on Rwandan building sites and farms

Moments are why the tools you see every day are shaped as they are. The long handle of a hoe (isuka) lets a farmer turn heavy soil with a smaller force. A builder's wheelbarrow carries sand and bricks up the hill with the heavy load close to the wheel. A bottle opener, a pair of pliers and a door handle all put the force far from the pivot. Try the PhET: Balancing Act (version 1.3.5) simulation: place bricks on a plank, predict whether it will tip or balance, then check.

How moments work: spanners and see-saws (Cognito). Watch the calculation moment = force × distance and how the principle of moments balances a see-saw.

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Practice

A force of 25 N acts at right angles to a bar, 0.40 m from the pivot. Find its moment.
Explain why door handles are fixed on the edge of the door furthest from the hinges.
A child of weight 350 N sits 2.0 m from the pivot of a see-saw. Where must a child of weight 500 N sit to balance it?
A uniform metre rule is balanced at its 50 cm mark. A 3.0 N weight hangs at the 10 cm mark. Where must a 4.0 N weight hang to balance it?
A wheelbarrow's load of 800 N acts 0.30 m from the wheel's axle, and the handles are 1.2 m from the axle. What upward force at the handles lifts it?
A pupil pushes a door with 60 N, at right angles, 0.80 m from the hinges. (a) Find the moment. (b) What force would give the same moment pushed 0.20 m from the hinges?
Your friend says, "A longer spanner does not help, because you still push with the same arm." Evaluate this claim using the idea of moments.

Show answersHide answers

$\tau = 25 \times 0.40 = 10\,\mathrm{N\,m}$
The moment is $F \times d$ . A large distance from the pivot (the hinges) gives a large turning effect for a small force, so the door opens easily.
$500 \times d = 350 \times 2.0 = 700$ , so $d = \frac{700}{500} = 1.4\,\mathrm{m}$ on the other side.
The 3.0 N weight is 0.40 m from the pivot: $3.0 \times 0.40 = 1.2\,\mathrm{N\,m}$ . Then $4.0 \times d = 1.2$ , $d = 0.30\,\mathrm{m}$ : at the 80 cm mark.
$E \times 1.2 = 800 \times 0.30 = 240$ , so $E = 200\,\mathrm{N}$ .
(a) $60 \times 0.80 = 48\,\mathrm{N\,m}$ ; (b) $F = \frac{48}{0.20} = 240\,\mathrm{N}$ , four times as much.
The claim is wrong. The moment is force times distance: with the same push, a longer spanner gives a bigger distance and so a bigger moment on the nut. For the same moment, a longer spanner needs a smaller force.

Key points

The moment of a force is $\tau = F \times d$ , with $d$ the perpendicular distance from the pivot; unit N m.
Principle of moments: in equilibrium, clockwise moments = anticlockwise moments.
Equilibrium needs zero resultant force AND zero resultant moment.
Levers (see-saws, wheelbarrows, spanners, hoes) use a large distance to give a large moment from a small force.

Sources and credits

9.2: The Second Condition for Equilibrium, from College Physics (1st edition) by Paul Peter Urone and Roger Hinrichs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Photo: Up and down.jpg by Roberto Picco, Wikimedia Commons, licensed under CC BY 2.0.
Photo: Two wheelbarrows.jpg by Oto Zapletal, Wikimedia Commons, licensed under CC BY 4.0.
Balancing Act (version 1.3.5). Simulation by PhET Interactive Simulations, University of Colorado Boulder, licensed under CC BY 4.0 (https://phet.colorado.edu)
Video: GCSE Physics - How Moments Work - Spanners and Seesaws, Cognito, embedded from YouTube.

The worked examples, diagrams, Rwandan applications and practice exercises are written by Ganzaa.

1Why is the handle far from the hinges?

2Experiment: balancing a metre rule

3The see-saw

4Levers on Rwandan building sites and farms

Why is the handle far from the hinges?

Experiment: balancing a metre rule

The see-saw

Levers on Rwandan building sites and farms