Forces and acrobats

The Tombolas are a circus act. The bottom row support the rest. They are supporting the three extra bodies. They must push upwards more than normal.

Pile up three large cushions. The shape of the cushions changes. What is happening to the one at the bottom?

Both situations involve things in equilibrium. The forces acting down on the cushions and on the Tombola brothers are each being balanced by a force acting upwards. We might call this upward force a support force.

But what about the ground under Papa Tombola's feet? It is also being squeezed and is supporting him (and his sons). The floor under the cushions is doing the same job. How can the ground exert a force upwards?

Compression forces

If you look closely at the ground it will be slightly distorted, dipping a little due to the force acting on it. We can explain these surface support forces by thinking of the particles which make up the surface. This takes some imagination.

These sketches show how we might imagine the particles in the floor. A simple model is to consider the atoms in the floor being connected by spring-like bonds.

It is rather like a tiny mattress structure. When a force acts downwards on the floor, the floor distorts slightly and the floor structure becomes compressed. The reaction to being compressed provides an upward force to support the gravity force and this upward force is often referred to as the reaction force from the surface (the floor in this case). It is an example of a compression force: something is compressed and exerts a force as a result of this compression.

This is how we can think of a solid inanimate object providing an upward supporting force. This upward supporting force is our first example of a contact force, where one object exerts a force on another object through being in direct contact with it (the floor pushes up directly on the Tombolas' feet).

Since the distortion does not have to be made by pushing downwards (you could push sideways by learning on a wall) a better general name for such forces is compression forces. Expect such forces wherever one object is in contact with its surrounding in such a way as to compresses the solids in the surroundings. Compression forces are drawn from the surface of the object, to remind you of the warp or distortion that caused the push.

Tension forces

When you lie on a bed the gravity force acting on you squashes the springs and they react, providing an upward force to support you. The mattress is a person-size model of what is happening at a microscopic level under the bottom row Tombola's feet. Forces that tend to squash objects are called compression forces. Forces that tend to extend objects are called tension forces.

If you extend a metal wire, perhaps a guitar string, you can feel the effect of the tension forces between the particles in the wire. As the material extends, the particles separate and the attraction between them pulls them back together. We say the wire is in tension.

Expect tension forces wherever one object is in contact with its surrounding in such a way as to stretch the solids in the surroundings (for example, a climber hanging on a rope). Tension forces are drawn from the surface of the object, to remind you of the warp or distortion that caused the pull.

Both tension and compression forces arise when solids in the surroundings are distorted or warped. We might therefore group tension and compression forces as warp forces.

But what about objects surrounded by fluids (liquids and gases)? Read on.

Exerting forces; changing the shape of things

Civil engineers, dressmakers, dentists and fishermen are just some of the people for whom a knowledge of the strength and flexibility of materials is important. For pupils, stretching materials is an area of science where they can do some real investigating on everyday items, such as T-shirts, carrier bags and party balloons, and learn about controlling variables, data collection and resource management.

The forces evident in such systems are invariably in equilibrium. When hanging masses on a spring, gravity provides a downward force that is balanced by an upward force from the stretched material. We often say this upward force is due to the tension in the material. As the force of gravity increases, the tension increases – often accompanied by a change in shape of the material – it stretches. It is this change in shape that might be investigated. For example, what happens to the length of a rubber band as we hang more masses on it? Squashing as well as stretching can change shape. Loading masses on a piece of foam cushion material provides the basis for an interesting investigation.

Relationships between force and change in shape

The relationship between force and change in shape isn't always simple. For a spring the extension is directly proportional to the force applied up to a limit beyond which the spring is permanently deformed. The force / extension graph showing this behaviour is a straight line up to the elastic limit.

For an elastic band the pattern is more complex and the force-extension graph is a curve.

There is more potential for fruitful reasoning as pupils explore the subtleties of stretching behaviour and the resulting graph than in the cruder breaking limit investigation. The latter poses a greater safety concern with heavy objects falling to the ground near tender toes. Stretching without breaking is a process that can be repeated without having to replace springs or rubber bands constantly.

Floating and sinking

Floating and sinking are often the subject of study in science lessons. An object that is afloat is an object in equilibrium. To keep an object afloat there must be an upward force to balance the pull of gravity on the object. You can feel this upward force whenever you swim or try to submerge yourself in the bath. (This buoyancy force is sometimes called the upthrust.)

This buoyancy force is shown by a vertical arrow from the bottom of the object, since the origin of the force depends on interactions between the object and the particles bombarding its surface.

To explain the origin of this buoyancy force (which provides an upthrust) you need to think about the pressure in the surrounding fluid (liquid or gas), and the resultant forces acting on the surfaces of the object by the fluid environment. You'll also see why we chose to draw the force arrow from the bottom surface.

Expect buoyancy forces wherever one object is in contact with its surroundings in such a way that there will be particles of the fluid bouncing off the object's surface.

Origins of the buoyancy force

This is an account of the origins of the buoyancy force.

Both boats floating in the water are in equilibrium. Can you identify all the forces that are acting on the boats in the sketches?

One boat makes a bigger hole in the water than the other. Another way of saying nearly the same things is that one boat displaces more than another. Here that's because one is loaded more than the other. In another case it might simply be that the boats are made out of different materials, so one boat is heavier than the other.

Forces in equilibrium

The gravitational force on an object (the pull of the Earth) is the one force you can always rely on to be present. If this were the only force acting on a floating object it would sink. To keep it stationary on the water there has to be another force, one acting upwards to balance the pull of the Earth. This is the buoyancy force. It is sometimes also called the upthrust of the water. This buoyancy forces supports the boat just as the table supported the teapot. The boat is an object in equilibrium, acted on by two forces.

The Plimsoll line

How big a hole is made in the water is important. You can overload boats. Cargo ships were regularly overloaded, and many seafaring nations were in the habit of painting marks on the sides of their ships to show just how much cargo you could load. The more you loaded on, the larger the hole in the water, and the lower the boat floated. This was standardised as a result of the work of Samuel Plimsoll, a British MP. He introduced several lines, to allow for journeys to different locations, in different seas and at different times of year. Although the boats were always displacing water, it was not always the same water.

And that's an important clue, as buoyancy forces are exerted by all fluids, including air.

Making holes in water – the larger the hole, the greater the force

Buoyancy then appears to be a variable force. One way to get a visceral feel for this force is to take a cylindrical plastic bottle and a basin full of water. Put the top on the bottle and depress the bottle into the water – and notice that it feels just like compressing a spring, or a piece of foam. The more you push the bottle into the water, the greater the support force exerted by the water. Another way of saying the same thing: the greater the hole you make in the water, the greater the buoyancy force exerted by the water.

Now switch bottles: change to a bottle which has four 1 cm holes punched in the sides at the bottom. Unscrew and remove the lid. Now when you push the bottle into the water you will feel no buoyancy force exerted – you're not making a hole in the water. You can push and pull this bottle up and down, no holes are made in the water, and so no buoyancy force is exerted by the water, and no buoyancy force acts on the bottle.

Back to the original bottle. Fill it about 1/3 full with water (you may wish to use food dye in the water so that you can see the water level inside the bottle clothes). The bottle will of course be heavier. Again push the bottle into the water. Now you should find that you don't have to push the bottle until it's already a short way into the water. In fact you'll now have to pull up to get it out of the water.

Where's the transition between pushing down and pulling up – how does the depth of the hole made in the water compare with the water level inside? This transition is of course when the buoyancy force provided by the water is exactly equal to the force of gravity acting on the water inside the bottle. The bottle is in equilibrium at transition. Repeat with different quantities of water inside the bottle to check your idea.

We hope you spotted that the buoyancy force acting on the bottle is exactly equal to the force of gravity acting on the water in the bottle when the water level outside the bottle is equal to the water level inside the bottle. In other words exactly equal to the volume of water you had inside the bottle. The buoyancy force exerted is equal to the force of gravity acting on the water displaced – the water pushed out of the way by the inserted bottle.

Density and floating

Why do some objects float and others sink? We've just noticed that the buoyancy force exerted is equal to the gravity force acting on the water displaced in the bottle-pushing experiment. Now maybe we could replace the material inside the bottle with something other than water. If the replacement makes the bottle heavier than the water, the bottle will make a larger hole to compensate, if lighter, than a smaller hole. If the hole the bottle makes is not large enough to compensate for the added pull of gravity, then the bottle will sink.

But we could forget the bottle, and move on to materials. Now the volume of the material will set the size of the hole made, and so the buoyancy force, whilst the mass of the material will set the pull of gravity.

A challenge

Here is a challenge for you. Consider the list of objects below and decide which ones will sink and which will float when placed in water.

• 3 kilogram block of iron
• 10,000 kilogram steel hull car ferry
• 0.02 kilogram piece of iron
• 10,000 kilogram wooden hull car ferry
• 500 kilogram block of wood
• 5 gram brass screw
• 10 kilogram piece of wood
• 5 gram plastic clothes peg
• 0.5 kilogram football
• 3 kilogram block of ice

As a general rule, for solid objects, everything made of wood floats and everything made of a metal sinks.

The particles in wood are not so closely packed. There is more space in between the particles and fibres, which gives wood its low density. Metals have their particles closely packed such that there is little space inside. This makes metals denser than either wood or water. The volume of water displaced (the size of the hole made) is small compared to the pull of gravity for high density materials.

We think it's helpful to imagine weighing a set of blocks of materials, all the same size. The blocks that weigh less than the same volume of water will float. The stuff from which they are made is not so densely packed. The blocks that are heavier than the same volume of water will sink. They have a densely packed structure.

A football floats because it contains plenty of air. Air is less dense than water. A football is a combination of a plastic outer shell and air interior. Overall the football is not very dense. For the same reason, steel boats float because their interiors contain so much air that their average density is less than that of water. So it is not strictly true to say that all metal objects sink. If they are shaped so as to contain air they may float: it all depends on the average density.

For solid objects (not shaped to trap air or another low-density material), the rule for floating is that an object will float in water if its density is less than that of water.

This is a reliable rule of thumb – but it's not an explanation.

So floating and sinking are all about the close or loose packing of particles – a property which can be measured by finding the density of a material.

Here you can find out more about density

To appreciate the idea of density just imagine you can look deep inside a material. If the particles are packed closely together there is a lot of mass in a small volume.

If, on the other hand, there is a lot of space between the particles, the material will be less compact and will not be so dense. This relationship between the mass of an object and the volume it takes up is the key to appreciating density. In the real world, the mass of the particles themselves might also be different and this will influence the density of the material. Some materials, for example honeycomb, also have large spaces in their structure. This will tend to reduce their density.

Anyone who has moved boxes when setting up a new home will tell you that some boxes contain many more items, densely packed, than others. The box with all of the books in it (densely packed) is the one to avoid carrying! Delicate glassware will have lots of air between the items and is easier to handle.

Explaining buoyancy

Back to boat: the larger the hole it made in the water, the greater the support force – the buoyancy force.

The physical origin of the buoyancy force is a challenge to explain. It involves appreciating that the pressure in a fluid increases as you go deeper. For water, just think about deep-sea divers and the special suits they have to wear to withstand the water pressures of the deep sea.

This extra pressure arises from the increased density, which leads to more bombardment of the bottom surface, as compared to the top surface. This extra pressure on the bottom of the object is the source of the upward force on the object. So whatever the force of gravity acting on the object, if it floats we can be confident that a force equal in size to the force of gravity acting on the object is pushing upwards. This is the buoyancy force. Let's look at that line of reasoning using diagrams, focussing on the boat as an example.

First simplify the boat to a box.

Then look at how the pressure changes with depth.

We can re-imagine the increasing depth of water as increasing pressures at different depths – caused by increasing bombardment by the particles of the fluid.

This bombardment leads to forces acting on the sides and the bottom of the model of the boat.

These forces add, and the vertical supporting forces balance the pull of gravity on the boat.

Buoyancy forces on fully-immersed objects

What about sinkers? The upward force doesn't disappear just because an object sinks. There are still more particles in each cubic metre at the bottom of the object than at the top. So there is still more bombardment, and so more pressure, at the bottom than at the top. You will notice that objects sink slowly, more slowly in water than in air. In such cases, the buoyancy force is still there pushing up on the object, but it is just not big enough to balance the gravity force.

Add a lid to the model of the boat and submerge the model. As the depth increases the vertical forces due to bombardment on the top and bottom surfaces both increase, but the difference stays the same. The buoyancy force, which is the result of this difference, also stays the same.

So there are still two forces acting but they are not balanced. The force of gravity acting on the object is the greater force and so the object sinks. For all objects denser than the surrounding fluid, as they sink the buoyancy force pushing on them is less than their force of gravity acting on the objects. The resultant force is downwards: the buoyancy force is smaller than the gravity force and the object sinks. For all objects less dense than the surrounding fluid, as they sink the buoyancy force pushing on them is greater than the gravity force. The resultant force is upwards: the buoyancy force is greater than the gravity force and the object rises.

For a floating boat, the bombardment on the underside is enough to balance the pull of gravity. If you get into the boat, the force of gravity acting on the boat and you increases, and more of the boat is immersed in the fluid. There is a greater rate of bombardment than before you got in (the bottom of the boat is further down in the fluid and there are more particles in each cubic metre to do the bombarding). If this new level of bombardment is enough to support the force of gravity acting on both you and the boat then you still float. If not, it sinks.

Describing friction

When two surfaces are in contact there is a force acting on each surface that acts in a direction to stop them moving past one another. This is often said to be due to friction, but there are two possibilities: movement, or no movement.

Rough surfaces have more friction than smooth surfaces and liquids such as oil or water are sometimes used as lubricants to reduce the effect of friction. There is no mystery behind friction. It acts on objects at the surfaces so as to prevent or reduce movement between the surfaces. When friction prevents sliding there is grip, when sliding is reduced there is slip. There is enough of a difference between these two that we suggest that you distinguish between grip forces and slip forces.

Grip forces are often a good thing. Without grip we couldn't walk anywhere. When we walk the grip force between our shoes and the ground enables us to push against the ground. On an icy surface walking is much harder because there is less almost no grip force.

Slip forces are useful in reducing existing motion. Without them the options for stopping a bicycle would be limited.

For a simple explanation of how grip or slip forces happen, imagine two surfaces at a microscopic level. All surfaces are full of imperfections. Nothing is totally smooth. When these imperfections catch on each other they act to prevent or reduce movement. Just think about rubbing two sandpaper surfaces together.

Friction through forces spectacles

Through forces spectacles a grip force or slip force is shown by drawing in a force arrow, parallel to the surface resting on the rough solid surroundings, to remind you of the origins of the force on the object. In the example below, the force acting on the box is shown by an arrow along the bottom surface of the box in the opposite direction to the intended motion.

A drag force

When objects move through a fluid (a liquid or a gas) their progress is opposed by a force which acts on the object and in the opposite direction to the movement. We call this a drag force and it acts just like a slip force. A drag force is not so much a rough, rubbing effect as a brushing-by effect. Air resistance is a drag force. Swimmers and high-divers experience a drag force when they try to move through water. Drag arises from wave making and the viscous movement of the particles of the fluid around the moving object. The mechanisms are complex: but identifying the force is not. If an object is moving through one or more fluids then there will be a drag force.

An interesting feature of drag forces is that they become greater as the speed of movement increases. Drag forces can therefore influence motion, either reducing the speed of an object in the case of a bullet or preventing an object from speeding up further in the case of a sky diver.

Through forces spectacles a drag force is shown by drawing in a force arrow, with its tip at the surface where the fluid has the greatest effect, to remind you of the origins of the force on the object.

Factors affecting drag

When trying to reduce drag, shape and speed are the two major factors.

After you have managed to do your best with these then move on to experimenting with the surface of the object to further reduce the drag. The aim is to disturb the fluid as little as possible as the object moves through it – so achieving smooth streamlines. You don't want any abrupt changes in the motion of the fluid past the surface of the object.

Swimming, cycling and motorcar racing are all activities where (frictional) drag forces are a hindrance. Modern science has helped us go faster by reducing the effect of drag forces, often studying the movement of animals for inspiration. Even low-tech strategies like shaving hair off the head and legs help swimmers to streamline their shape and so go faster.

Relating drag to speed

Here is an account of how the varying force of drag comes into equilibrium with the force of gravity to speed falling objects up to their terminal speed.

As a general rule, the faster you move the greater is the drag force acting against you. Objects falling through air experience a drag force that prevents them from speeding up too quickly. Eventually this drag force will balance the downward pull due to gravity resulting in no more speeding up – the object just continues to fall at a steady speed.

Skydivers experience this effect. As they fall they speed up, but the faster they go the greater is the drag caused by the air. Eventually they will reach a speed at which the size of the drag force equals the size of the pull of gravity. This is called the terminal speed (we'd recommend avoiding terminal velocity, unless you really do want to introduce the distinction between speed and velocity here).

It is interesting to appreciate that when this happens the falling skydiver is in fact in equilibrium. Can you make an argument to explain why?

A summary of contact forces

There are three kinds of contact forces that can support an object.

Warp forces can be found wherever a solid is distorted by an object:

• Add a compression force exerted by a neighbouring solid acting on the object if that solid is compressed by the object.
• Add a tension force exerted by a neighbouring solid acting on the object if that solid is stretched by the object.

You might, for teaching purposes, combine these two and call them warp forces – with the forensic clue that if a solid in contact with the object is stretched or squeezed then you can add an arrow labelled warp force.

• Add a buoyancy force if the object is partially or wholly immersed in a fluid.

Frictional forces of three kinds can be found at the surfaces of the object when it moves, or makes to move, past other particles in its environment.

• If the environmental particles are a solid and no movement occurs, add an arrow at the contacting surface and label it grip force.
• If the environmental particles are a solid and movement occurs, add an arrow at the contacting surface and label it slip force.
• If the environmental particles are a liquid and movement occurs, add an arrow at the most significant surface and label it drag force.

You might, for teaching purposes, combine these three and call them frictional forces – but we'd not recommend that as it obscures the very different reasons for adding the arrows.