FORCES 7. Pressure and upthrust in liquids, why do objects float or sink?, variation of atmospheric pressure with height

Doc Brown's Physics Revision Notes

Suitable for GCSE/IGCSE Physics/Science courses or their equivalent

What causes pressure in the atmosphere?

Why do objects float or sink in fluids?

What causes atmospheric pressure?

Why does atmospheric pressure vary with height?

Upthrust - why do some objects float in a liquid and others sink?

When an object is partially or wholly submerged in a fluid it experiences a force from all directions due to the pressure of the fluid (gas or liquid) - this force is known as the upthrust.

This force acts at right angles to the whole of the surface of an object.

Because pressure increases with depth, the object experiences a greater force at its bottom compared to the top.

The net resulting force on the object, acting in an upward direction, is called the upthrust.

diagram needed

The upthrust force is equal to the weight of the fluid displaced by the object.

The displaced fluid equals the volume of the object that is actually in the fluid - partially or completely sunk in the fluid.

If an object floats the weight of the fluid displaced equals the weight of the object floating.

If an object sinks its weight is greater than the weight of fluid displaced.

Water has a density of ~1000 kg/m3 and air has a density of ~1.2 kg/m3.

If the fluid upthrust is greater or equal to the weight of the object, then the object rises or floats.

This is why a helium balloon floats and rises in air. Helium is less dense than air (~0.18 kg/m3).

Our average density must be less than water, because we can 'float'.

Porous balsa wood (density 340 kg/m3) floats on water.

If the weight of the object is greater than the upthrust, then the object sinks.

This is why a solid steel object (density ~8000kg/m3) like a fork, will sink in water. Iron is much more dense than water.

Brick and concrete objects have densities of ~2200 kg/m3, so they sink in water.

SO, the deciding factor is the comparison of the densities of the object and the fluid.

If the density of the object is greater than that of the fluid, it will sink.

An object that is more dense that the fluid it is placed in, cannot displace enough fluid to equal its weight.

This means the weight of the object is always larger than the upthrust and so it cannot float and will sink.

But beware, the shape of the object can mean high density materials can float.

Iron is nearly eight times more dense than water, but shape it into a boat and it floats.

This is because the boat shape allows the displacement of water equal to the weight of the iron ship, so it floats!

If the density of the object is equal to, or less than, the density of the fluid it is immersed in, then the object will float.

An object that is less dense than the fluid it is put in, weighs less than the volume of fluid equal to its own volume.

Consequently, the object can only displace a volume of fluid equal to its own weight before it can be completely submerged - so it floats because the objects weight is equal to the upthrust.

You are used to the idea of objects floating in water, but helium balloons float in air!

The weight of the helium balloon is far less than the weight of the volume of air it displaces.

Therefore the upthrust from the air is greater than the weight of the balloon which will rise - as you will have observed as you see a 'freed' helium balloon rise high into the atmosphere.

Helium balloons are used by weather scientists and weather forecasters to get information on the weather conditions at high altitudes.

Since atmospheric pressure decreases with height, the imbalance between the balloon's internal and external pressures results in the helium balloon expanding. Eventually

Density is very important property to know about a material

e.g. if the average density of an object is less than that of water (~1000 kg/m3) it floats

if the average density of an object is more than that of water it sinks!

In general: if the object has an average density < fluid it floats and if the average density of the object is > fluid it sinks.

Note the phrase 'average density' - this is one way of explaining e.g. why iron boats float!

Because of the shape of the boat, the average density of the boat (steel + contents + air) is less than water.

Therefore the ship can displace a volume of water equal to its weight, without sinking and can therefore float.

This must be appreciated using explanation is to do with upthrust and displaced of fluid as described above.

Of course, if the boat develops a leak, the less dense air is displaced by water and when the ship fills up, due to the steel, the average density is greater than water and the ship sinks!

Some 'kitchen sink' experiments on density!

A The apple has a density of ~900 kg/m3 and floats on water, of density 1000 kg/m3

The upthrust of the water is equal to the weight of the apple, the forces balance and the apple floats.

The weight of the water displaced by the apple equals the weight of the apple.

B The potato has a density greater than water and sinks, so we have no idea what its density might be.

The upthrust of the water is less than the weight of the potato, the forces are unbalanced and the potato sinks.

The potato is unable to displace enough water to equal its weight, so it can't float.

The weight of the water displaced by the potato does NOT equals the weight of the potato, so it sinks.

C The block of wood has density of ~500 kg/m3 and floats (it toppled over!) - its density is half that of water and is only half submerged - can you reason out why it is only half submerged?

The upthrust of the water is equal to the weight of the block, the forces balance and the block floats.

D A block of wood of density ~1000 kg/m3, the same density as water, so it floats with its upper surface coincident with the surface of water. It couldn't be any more immersed in the water without sinking!

E Another block of wood of density ~600 kg/m3, so more of it is immersed in water than the other less dense bloc of wood above.

F I made a small 'barrel' of ice and placed in the beaker of water. Since ice has a density of 917 kg/m3, it floats on water. You can just about see that about 10% of the 'mini-iceberg' is above the surface of the water.

Its the 90% of an iceberg below the surface that is the main danger to ships (the Titanic!), not the 10% you can see!

As far as I know, ice is the only solid form of a substance that is less dense than its liquid form.

The molecules of ice form a quite widely spaced crystal lattice so that on average the molecules are further apart than they are in the liquid, despite the contrast between the ordered structure of ice and than randomness of liquid water (if you do A level chemistry you go into the details of hydrogen bonding in water - quite interesting!)

Other cases of floating versus sinking   (noting the density of water is 1000 kg/m3)

Iron nails will float on the much more dense liquid metal mercury.

Group 1 metals: Lithium - density 535 kg/m3 floats on water, caesium - density 1870 kg/m3, sinks in water.

By changing the upthrust submarines can sink below the surface of water or rise back to the surface.

Sea water can be pumped into tanks to increase the weight of the submarine to that greater than the weight of the volume water it displaces i.e. the weight of the submarine is greater than the upthrust from the sea water.

Far less dense compressed air can be pumped in to the water tanks, displacing the water, and decreasing the weight of the submarine to less than the weight of the volume of sea water displaced by the submarine. The upthrust force is now greater than weight of the submarine which can rise to the surface.

By controlling the air and water levels in these buoyancy tanks you can stabilise the submarine at various depths.

You can also get material with a density greater than the fluid to float by changing its shape e.g.

You can build a ship of steel (density of ~8x water) that floats on water. The shape of the hull, filled with very low density air, allows the displacement of water equal to the weight of the ship. If you consider the whole of the volume of the ship, the average density will be less than that of water. Any individual steel plate of the ship will of course sink, as will the ship if it is holed and water flows in! diagram

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Atmospheric pressure and variation with height

The density of gases varies considerably with temperature and pressure.

Gases are very compressible because of the space between the particles.

You can squeeze the particles of a gas closer to together if a force is applied to them.

If you increase the temperature and the gas can expand, the density will decrease.

The atmosphere is a mixture of gases (mainly ~1/5th oxygen, 4/5th nitrogen) that surrounds the surface of the Earth.

It is relatively thin compared to the radius of the Earth but does stretch upwards for ~100 km, so there is quite a weight of air pushing down on us creating what we experience as atmospheric pressure.

However, the force acts in all directions so the internal pressure inside our body is the same as the external pressure beyond our skin. Therefore we do not experience this pressure, we are unaware of it and we don't change in size!

The graph on the left shows in principle how the atmospheric pressure varies with height above the Earth's surface (altitude

At the surface (height of zero km taken as sea level) it is normally close to an average of 101300 Pa (~101 kPa).

At the top of the world's highest mountain, Mount Everest in Nepal, the air is much thinner at ~8800 m above sea level. Here the pressure is only ~33000 Pa (~33 kPa) which is why breathing is much more difficult. Although your internal and external body pressures are equal, you take in less air-oxygen in each breath so all physical work is much harder than at sea level. The first mountaineers to reach the summit used cylinders of oxygen, but today's super-fit climbers can manage on just thin air! The local Sherpa's come from an ethnic mountain population that have evolved in several ways to cope with the local conditions e.g. their mitochondria are more efficient at using oxygen in respiration and blood flow in small blood vessels doesn't decrease as much as happens with non-Sherpa people.

The atmospheric pressure around us is due to the collision of air molecules on any surface AND the weight of the gas above you (note there are two contributions to atmospheric pressure).

So why does atmospheric pressure vary with height?

At very high altitudes there is little air, far few collisions, less weight of gas above and so the pressure tends towards zero Pa.

The greater the height/depth of a gas, the greater the weight of particles that gravity is pulling down, hence the increase in force per unit area at a particular level i.e. increase in pressure.

As you increase in height above the Earth' surface (increase in altitude) the atmospheric pressure decreases.

This is because the air is less dense and so less collisions can take place in a given volume AND there is less weight of molecules above a given altitude. Therefore the greatest atmospheric pressure will be less at the Earth's surface.

To express and explain the trend in another way:

The increase in pressure the nearer you are to the Earth's surface, is due to the greater density - hence more collisions AND the greater the weight of air above you - the greater force per unit area.

The weight of air above a certain height compresses the atmospheric gases below that level and compression means increase in pressure (ignoring any temperature differences).

Just as with liquid fluids discussed above, gases are fluids and the weight of them acting downwards creates a pressure in the same way.

At a given height above the Earth's surface, there is very little variation in the density and pressure of the atmosphere.

However, most weather systems are driven by regions of higher or lower pressure compared to the average atmospheric pressure at that height.

If you look at weather charts on the TV weather forecast you will see a 'high' (H) area with a number like 1029 by it, conversely, a 'low' (L) might have a number like 986 by it.

The average surface atmospheric pressure is ~1000 millibars (but don't worry about this unit, but 1 millibar = 0.1 kPa).

Forces revision notes index

FORCES 1. What are contact forces & non-contact forces?, scalar & vector quantities, free body force diagrams

FORCES 4. Elasticity and energy stored in a spring

FORCES 5. Turning forces and moments - from spanners to wheelbarrows and equilibrium situations

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