• ## 01 Before you go any furtherMa01PNnugget01 Introduction

### Closing the door

Get up, walk to the nearest door and open it. Not so hard was it? You can probably do it with your little finger. Now close it, but this time try pushing very close to the hinge. Much harder. This is no longer a task which you can easily do with your little finger!

In both cases you have exerted a force on the door to turn it. The end result is the same, yet the forces exerted are very different. In pushing closer to the hinges a much bigger force is needed to close the door.

Try this simple demonstration out with your class: Challenge the biggest and strongest pupils to close the door by pushing near the hinges!

By the time that they arrive in secondary school, all pupils have built up a wide experience of pushing and pulling objects to make them turn about some kind of pivot. For example, as young children play on a see-saw they shift around to make the beam balance without even thinking about it. If a young child finds it hard to open a door, you'll not see them making their next attempt by pushing closer to the hinge!

• ## 02 Describing turning events – close that door!Ma01PNnugget02 Exposition

### The door through forces spectacles

Let's think about closing a door. Put on your forces spectacles and think about the forces acting. The door is anchored along one edge, so there is an obvious pivot. As you push on the door, you provide a driving force to make the door move. The hinges (especially if not well oiled) provide a frictional counter force in the opposite direction. This counter force is added to by the drag as you push the door through the air. Remember the useful pattern, from the SPT: Motion topic:

• If the driving force is greater than the counter forces, an object will speed up.
• If the driving force equals the counter forces the object continues at a steady speed.

Here, and in our other examples, we'll try and keep the analysis simple by dealing with situations where the door spins at a constant rate, not speeding up or slowing down. So we'll keep to the same patterns as much as possible. But, for thinking about the door or anything else that spins, it is not enough just to check that the driving and counter forces balance each other. We must also take note of how far the forces are from the pivot to see that the door spins steadily.

The skills in identifying forces will be drawn on again. You can draw an abstract forces picture for the door, showing the pivot, and the forces correctly located with respect to that pivot.

Drawing diagrams such as this is a first step in analysing the world of levers, where forces act around pivots. If you apply your force very close to the pivot the turning effect (more on this shortly) produced is rather small. If you push farther away from the pivot the turning effect is bigger.

• ## 03 Levers everywhereMa01PNnugget03 Exposition

### Forces acting to cause turning

Once you start looking, you'll see forces causing turning everywhere. Take a look at this pair of examples and try to identify the forces acting to cause turning.

• A fridge door.
• Wash-basin taps.

You might then see how well you can identify levers in everyday situations. (Two to get you started: a crow bar and a bicycle pedal.)

• ## 04 The forces acting on leversMa01PNnugget04 Exposition

### Forces acting on levers

The simplest lever has three forces acting on it. Think about removing the metal cap from a bottle (maybe a beer bottle) with a bottle opener. You pull up on one end of the bottle opener, the bottle opener pivots on the middle of the metal cap and the lip of the opener forces the lid off.

The three forces at work on the lever are:

• The force of your hand as it pulls up on the handle of the lever: Sometimes called the effort applied to the lever.
• The force of the cap as it pulls down on the lip of the lever (this is the paired force of the lever pulling up on the cap): Sometimes called the load on the lever.
• The force of the corner of the lid as it acts downwards on the lever at the pivot.

With these three forces balanced, the lever stays still; it does not accelerate off, either upwards or downwards.

All three forces are linked and so to lift off a particularly well fastened cap, you need to pull up harder on the end of the opener (a bigger effort). As a result the lip of the opener will push up more on the lid, opening the bottle and the corner of the lid pulls down more (at the pivot).

In thinking about how levers work we are really just interested in those forces which cause turning and can therefore ignore the forces acting at the pivot.

• ## 05 Trading force and distanceMa01PNnugget05 Exposition

### A bottle opener

With the bottle opener you:

push down with a small force (your effort) … through a large distance.

This results in:

a large upward force of the lever on the cap … acting through a small distance.

The whole point of the bottle opener is to produce a big force from a small force, but the drawback is that you must push through a big distance. Devices such as this are sometimes called force multipliers.

• ## 06 Designing levers – turning effects and momentsMa01PNnugget06 Exposition

### Force and distance

We have seen from the previous examples of working with levers that the two quantities which need to be taken into account are the size of force acting and the length from the pivot.

More precisely we need to take account of:

• The sizes of the forces acting F1 and F2
• The perpendicular lengths between the line of action of each force and the pivot point L1 and L2

The line of action of a force is exactly what the name suggests: The direction or line along which the force acts. In the diagram, F1 acts along a line vertically upwards: so does F2 – but along a different line.

Both of the forces F1 and F2 produce a turning effect on the lever:

• F1 tends to turn the lever in a clockwise direction.
• F2 tends to turn the lever in an anti-clockwise direction.

The size of these turning effects can be calculated.

### Calculating turning effects

Here's a precise way of writing it out, so that every term is just a number:

turning effectnewton metre = perpendicular lengthmetre × forcenewton

You can also express it rather concisely as:

turning effect = perpendicular length × force

### The moment

The turning effect is called the moment of the force and is measured in newton metres.

For the example given above (given that the lever is balanced): clockwise moment = anti-clockwise moment, so L1 × F1 = L2 × F2.

This law of balancing actually follows from the principle of conservation of energy.

If you are interested in seeing how one bit of physics can be used to explain another, take a look at the following expansion nugget.

• ## 07 Lifting things with leversMa01PNnugget07 Expansion – lead me deeper

### Conservation of energy: trading force for height

Here we show how the required relationship between force and length to pivot to achieve a much lower lifting force is the consequence of the conservation of energy.

In episode 02 of the SPT: Energy topic we referred to a fundamental rule expressed by some people as: You can never get owt for nowt!. In physics this is expressed more formally as the principle of conservation of energy.

Let's think for a moment about a particular event which involves lifting something. To make this happen a minimum amount of energy must be shifted from the chemical store of your arm to the gravity store of the object in the Earth's field. This is connected to levers and doors – just follow the argument for the moment!

We can calculate the energy shifted here (see episode 04 of the SPT: Energy topic) using:energy shifted = force exerted × height

Where force exerted is the magnitude of the force needed to lift the object and height is the distance through which that force moves. You'll meet this relationship rather often, usually as: energy = force × distance

You can get the same job done in four stages, using one quarter of the force but compensating for this reduction by having to lift four times the height. This compensation leaves the energy shifted unchanged, but it does make the job easier since you need to exert a smaller force to lift each of the quarters.

Another way forward is to get this quadruple-length-lift over with in one smooth action by using a lever.

### Designing a lever for this task

But how can we set up this lever to lift the object by exerting just one quarter of the force? We know that the same quantity of energy must be shifted as with a direct lift (never getting anything for nothing). Then the design of the lever becomes very simple. To lift the object with a force which is one quarter of the force of gravity acting on the object, you need to use a lever where you push four times further from the pivot than the position of the object.

• ## 08 Starting out on calculationsMa01PNnugget08 Exposition

### Lines of action and forces

Describing the world using forces can be tricky, since you cannot see them.

For situations where levers are used you need to be able to correctly identify the forces acting on the lever and lengths to the line of action of those forces from the pivot point.

### Moments

Then you can calculate moments. Since lengths are measured in metre and forces in newton, be sure to give the moments units of metre newton (m N).

You should be able to show that the moments shown here are:

• 1 metre newton.
• 0.2 metre newton.
• 16 metre newton.

### Balanced levers

Once you have the idea that the clockwise moment is equal to the anti-clockwise moment, L1 × F1 = L2 × F2, you can tackle some simple calculations which allow you to find a missing force or length if you know values for the other three quantities.

Here are a few for you to try out, just to build confidence.

You should be able to show that the unlabelled forces are:

• 5 newton.
• 0.4 newton.
• 9 newton.
• ## 09 Thinking more widely – bent leversMa01PNnugget09 Expansion – tell me more

### Bent levers

This helps you to recognise levers at work in a wider range of contexts.

If you are careful in correctly identifying the length from the pivot to the line where the force is acting, you can design a wider variety of levers. It is not necessary for levers to be straight.

• ## 10 Various terms used for the turning effect of forcesMa01PNnugget10 Expansion – tell me more

### Other terms for the turning effect of forces

Moments are not the only words used to quantify the action of forces that cause turning. Read more here.

Moment is the only word that we choose to use here for the turning effect of a force acting around a pivot. You may also see the words torque and couple referred to. In fact engineers regularly use calibrated torque wrenches (specially designed spanners) to tighten bolts by just the right amount. Mountain bikes might typically have bolts which need to be tightened with moments of 4 metre newton to 12 metre newton. If you try to tighten beyond the setting of the torque wrench, it starts slipping.

The term torque might be reserved for those situations where the turning effect is transmitted by a shaft. Screws, bolts and drive shafts can all be described as providing torques.

A couple is simply a pair of moments. The forces and lengths are identical in magnitude, but act in opposite directions.

• ## 11 Screwdrivers are used for many things – be careful!Ma01PNnugget11 Exposition

### Screws or tins?

Explaining how screwdrivers work can be tricky because they are designed for one thing but often used for other jobs (especially flat-bladed screwdrivers). For example, the screwdriver might be used at one moment to tighten a screw in a kitchen cabinet and at the next to lever the lid off a tin of treacle. Driving in screws is made easier if the handle is made larger in diameter than the screw head. In this way the small force exerted by the hand on the handle becomes a large pair of forces acting on the slot in the screw head. This increase in force is traded for a decrease in the distance moved by the screw head compared to the hand. The screwdriver, used like this, is a force-multiplier.

If you doubt this try exerting a force on the screw with your hand, directly. If you still find the screw rather hard to drive in, choose a screwdriver with a thicker handle!

However, if you are using the screwdriver to lift up a lid and are still having difficulty, then choose a screwdriver with a longer shaft. You will find longer and shorter screwdrivers in toolkits – not for opening cans – but longer screwdrivers simply offer easier access to awkwardly placed screws (maybe deep inside a computer cabinet).

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• ## 12 Stability and topplingMa01PNnugget12 Expansion – tell me more

### Stability

Stability can be connected to the turning effects of forces. Here is how.

Thinking about the turning effect of a force can also shed light on why some things are stable and others not. Tilt a toy building brick along one of its long sides, then let go. How far can you tilt it before it does not fall back to its original position?

Try again with a longer side.

### Toppling

A simple and useful measure of stability is the maximum angle to which you can tip something with it returning to its initial state once you let go. If you tilt the object only a small amount, the force of gravity acting on the object is still turning it back to its original position. However, if you tilt beyond a certain angle, the force of gravity is now turning the object to a new position. The object will no longer fall back onto its base but will now fall onto its side.

Think about these situations for three vehicles viewed from end on. Which of the vehicles will tip over?

### Centre of mass

A helpful way of looking at this question is to say that things are stable so long as their centre of mass (through which the force of gravity acting on the object acts) lies within their footprint.

If an object is tilted and the line of action of the force of gravity falls outside its footprint, the object topples over.

So the centre of mass of an object is connected to the turning effect of forces.

Try pushing a book across a smooth table without making the book spin. Did you succeed? Then you are pointing at the centre of mass. Try again pushing along an edge at right angles to the first. If you manage another successful pointing act, then the centre of mass of the book is on the line where these two intersect. It has to be a line because the book is three-dimensional, and you'd need to make a measurement along the third axis to choose where along the line the point that is the centre of mass lies.

This centre of mass is a very special place for the book. Earlier on in this topic you worked with forces acting along one line only. If the forces make the object move without spinning, then this line must run through the centre of mass. So when we simplify the world to forces acting on point particles, the points must be at the centre of mass of the objects that they represent.

The centre of mass of the book is where the particle that represents the whole book collapsed to a single point would be.

• ## 13 Describing and using leversMa01PNnugget13 Summary

### Force, energy and moment

To set a believer you need to choose a pivot. Once you had done this, you are able to measure the distances from the forces exerted to the pivot, and the distances that those forces move when the lever rotates. These two measures allow you to calculate both moment and energy. Energy conservation is more fundamental.

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