Upthrust
Forces and Motion | Properties of Matter

Forces: floating and sinking

Physics Narrative for 11-14 Supporting Physics Teaching

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.

Upthrust
is a special case of Force
appears in the relation Upthrust = Weight of Displaced Liquid
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