Pressure · Model
Upthrust, floating and sinking
Drop a steel bolt in a bucket and it goes straight to the bottom. A steel ship the length of a street floats. Same steel.
Start here
Push a beach ball under and it fights you.
Hold one under the surface of a pool. You can feel the water shoving it back up the whole time, and the further under you push it the harder the shove gets. Let go and it leaves the water.
Where does that upward shove come from?
It comes from the pressure being different at different depths. The bottom of the ball is deeper than the top, so the water pushes up on the bottom harder than it pushes down on the top. Those two pushes do not cancel, and what is left over is a force upwards. It has a name: upthrust.
Upthrust is the upward force a liquid or gas exerts on anything in it, and it exists because pressure increases with depth. It is equal to the weight of the liquid the object pushes out of the way. An object floats when it can push aside a weight of liquid equal to its own weight, so that upthrust and weight balance; it sinks when even fully under it cannot push aside that much, so the weight wins.
At the bench · five blocks, one tank
Every block is one litre. Only the weight changes.
Change a control to begin
One litre of water weighs 10 N, so a block pushed fully under can never get more than that much upthrust. Pick a block. Then try holding it right under.
Commit first. Two one-litre blocks, one of pine and one of steel, are both held completely under the water. Which gets the bigger upthrust?
The block
Your hand
Weight in air
—
Upthrust now
—
On a spring balance in the water
—
What it does
—
The relationship · a beam, not a triangle
It floats when the upthrust equals the weight
R = W − U
Two arrows the same length balance.
The longer one wins, and the leftover is what moves it.
Worked example · one step at a time
A stone weighs 12 N in air. Hanging fully under water, the spring balance reads 9 N. What is the upthrust?
Step 0 of 5
Convert
12 N stays 12 N · 9 N stays 9 N
Both readings come off the same spring balance in newtons, so there is nothing to convert.
Formula
upthrust = weight in air − reading in water
The balance loses exactly what the water is supporting.
Insert
upthrust = 12 N − 9 N
Both readings come off the same spring balance.
Fine-tune
12 − 9 = 3
Newtons take away newtons, so the answer is in newtons.
Answer
upthrust = 3 N upwards
Three newtons up — and the stone still sinks, because 9 N is left over pulling down.
Worked example · one step at a time
A metal cube of mass 1.2 kg hangs from a balance. Fully under water the balance reads 8.0 N. What is the upthrust?
Step 0 of 5
Convert
1.2 kg × 10 N/kg = 12 N
The subtraction needs two forces, and a mass is not a force — weight in newtons is mass in kilograms × 10 N/kg.
Formula
upthrust = weight in air − reading in water
The balance loses exactly what the water is supporting.
Insert
upthrust = 12 N − 8.0 N
The converted weight goes in. The 1.2 kg never does.
Fine-tune
12 − 8.0 = 4
Newtons take away newtons, so the answer is in newtons.
Answer
upthrust = 4 N upwards
Subtract 8.0 from 1.2 and the upthrust comes out negative, which no upthrust ever is.
Your turn · the same five steps
Your block: pine, 5 N, with 5 N of upthrust on it.
Write all five lines before you check. The numbers are the ones your own tank is showing.
The five lines, marked
Convert
5 N stays 5 N · 5 N stays 5 N
Both are already forces in newtons, so there is nothing to convert.
Formula
left over = weight − upthrust
A difference between two opposite forces, so a beam rather than a triangle.
Insert
left over = 5 N − 5 N
Both figures come off the bench above, bigger one first.
Fine-tune
5 − 5 = 0
They cancel exactly, which is what floating is.
Answer
nothing left over — it floats
Upthrust and weight are equal, so nothing is left over.
The five lines give nothing left over — it floats, and the two arrows on the bench are drawn to match.
A sinker of mass 0.60 kg hangs from a balance. Fully under water the balance reads 4.5 N. What is the upthrust?
This one needs the Convert line to do some work.
The five lines, marked
Convert
0.60 kg × 10 N/kg = 6.0 N
A mass is not a force. Weight in newtons is mass in kilograms × 10 N/kg.
Formula
upthrust = weight in air − reading in water
The balance loses exactly what the water is supporting.
Insert
upthrust = 6.0 N − 4.5 N
The converted weight goes in. The 0.60 kg never does.
Fine-tune
6.0 − 4.5 = 1.5
Newtons take away newtons, so the answer is in newtons.
Answer
upthrust = 1.5 N upwards
Subtract 4.5 from 0.60 and the upthrust comes out negative.
The five lines give 1.5 N upwards. The whole question turned on the first one.
Key fact
Upthrust is the upward force from a liquid or gas, and it equals the weight of what the object pushes out of the way. It floats when the upthrust can match its weight, and sinks when even fully under it cannot.
Think again
“Heavy things sink and light things float.”
A cargo ship weighs two hundred million newtons and floats; a steel bolt weighs half a newton and sinks. Weight alone decides nothing. What decides it is the weight of the object compared with the weight of water it can push out of the way — and a hull shaped like a hull pushes aside thousands of tonnes of sea, while a bolt can only push aside a bolt's worth. Squash that same ship into a solid cube and it goes straight down, because now it pushes aside almost nothing.
“Only things that float get upthrust.”
Everything in a liquid gets upthrust, sinkers included. Hang a stone from a spring balance and lower it into water: the reading drops, and the amount it drops by is the upthrust. The stone still sinks, because the upthrust is not enough to match its weight — but it is definitely there, which is why a heavy rock is easier to lift while it is still under the water and suddenly feels heavier as it breaks the surface.
Mastery ladder
Not started yet.
Rungs 3 and 4 you mark yourself.
Rung 1 · Calculate
A metal cube weighs 45 N in air. Hanging completely under water, the spring balance reads 37 N. What is the upthrust on it?
Rung 2 · The one that catches people
A steel bolt sinks. A steel ship floats. Which statement explains it?
Rung 3 · Explain
The steel from a ship is melted down and cast into one solid block, which sinks. Explain why the ship floated and the block does not, using upthrust and the water pushed out of the way.
Rung 4 · Take it somewhere new
A submarine dives by letting sea water into its ballast tanks and surfaces by blowing it out with compressed air. Explain both, and say what stays the same throughout.
Key note
Because pressure grows with depth, a liquid pushes up on the bottom of an object harder than it pushes down on the top, and the difference is upthrust. Upthrust equals the weight of the liquid pushed out of the way. Floating is the balanced case: upthrust equals weight, nothing is left over. Sinking is the unbalanced one: even fully submerged the upthrust is smaller than the weight, and the leftover force takes it down.
Going further
A submarine uses the one thing the blocks on the bench could not: it changes its own weight without changing its shape. Flood the ballast tanks and the weight goes up while the upthrust stays exactly the same, so it dives; blow the water out with compressed air and the weight drops below the upthrust, so it rises. Hovering at a set depth means matching the two to within a few newtons, which is why submarines trim constantly and why a change in the saltiness of the water is something a crew has to notice.
The same balance is written on the side of every merchant ship as the Plimsoll line — a set of marks showing how deep the hull may legally sit in fresh water, in salt water, in summer and in winter, because each of those changes the weight of water the hull pushes aside. Gases do it too: a hot-air balloon floats in air for exactly the reason a cork floats in water, since hot air inside the envelope weighs less than the cold air it pushes out of the way. And the whole idea is old. Archimedes worked out the rule about the weight of the fluid displaced more than two thousand years ago, and shipbuilders still use it unchanged.
Before this lesson
Connects to
At GCSE this becomes
- Upthrust from the pressure difference across a submerged object, density as the test for floating, and Archimedes' principle written as an equation.
Where to next
Ask Mr Badmus AI
Got something you cannot decide will float or sink?
The tank is a teaching model. Every block is exactly one litre and water is taken as 1000 kg per cubic metre, so one litre of water weighs 10 N with weight as mass × 10 N/kg; the block densities are typical values and cork in particular varies a great deal. The weight arrow is drawn at a fixed length and the upthrust arrow in proportion to it, so the two can be compared with each other but not measured off the page; ratios beyond about two and a half are clipped to fit. How much of a floating block sits below the surface is calculated from the weights alone, ignoring the shape of the block and the skin of the water.
Lesson content © MrBadmusAI.