It is a hot afternoon at a roadside stall. A glass of nimbu pani is filled to the very brim, with a cube of ice floating in it. The ice stands a little above the rim. Everyone who sees it expects the glass to spill as the ice melts.
An ice cube floats in a glass of water filled exactly to the brim. Part of the ice stands above the rim. What happens to the water level as the ice melts?
In the figure above, the ice cube has sides of \(5\,\text{cm}\) and floats \(4.5\,\text{cm}\) deep. It displaces \(112.5\,\text{g}\) of water, its own mass. When it has melted, those \(112.5\,\text{g}\) of water fill \(112.5\,\text{cm}^3\), exactly the space the ice pushed aside. So not a drop spills, and the level never moves.
Now choose the chip with a stone frozen inside the ice. While frozen in, the stone is floated by the ice, so it displaces its own weight of water. When the ice has melted enough, the stone sinks. On the bottom it displaces only its own volume, which is less than its weight of water. The level falls. In the figure the stone has a mass of \(1.3\,\text{g}\) and a volume of \(0.52\,\text{cm}^3\), so the displaced volume falls by \(0.8\,\text{cm}^3\). In a glass \(10\,\text{cm}\) square, the level falls by less than a tenth of a millimetre, drawn much larger.
Ice in the sea is different. Ice of \(900\,\text{g}\) floating in sea water displaces \(900\,\text{g}\) of sea water, which fills \(878\,\text{cm}^3\). When it melts it becomes fresh water, which is less dense and fills \(900\,\text{cm}^3\). So floating sea ice raises the level very slightly as it melts. Ice melting on land, which was never floating, raises the sea by far more.
A glass is filled to the brim with water, and a piece of ice floats in it. In which cases does the level stay exactly at the brim when all the ice melts?
Floated or sunk
The ice gives a rule for every case of this kind. The level of water in a tank depends only on the total volume the bodies in it displace. A body that floats, by itself or through what it is tied to, displaces its own weight of water. A body resting on the bottom displaces only its own volume. For a body denser than water, that is less.
Take the block, the metal and the tank of Fig 4.1. Find the extra volume of the block under water when the metal is on top and when it hangs below. Then find how far the level falls when the metal is dropped.
The metal weighs \(8\,\text{N}\), and its volume is \(0.8/8000 = 10^{-4}\,\text{m}^3\). Under water it feels a buoyant force of \(1\,\text{N}\).
On top, the block must displace an extra \(8\,\text{N}\) of water. Hung below, the metal displaces its own \(10^{-4}\,\text{m}^3\), and the string pulls the block down with \(8 - 1 = 7\,\text{N}\).
On top the block's extra volume under water is \(8 \times 10^{-4}\,\text{m}^3\). Hung below it is \(7 \times 10^{-4}\,\text{m}^3\), plus the metal's own \(10^{-4}\,\text{m}^3\). The total is \(8 \times 10^{-4}\,\text{m}^3\) both ways, so the level is the same.
Dropped, the metal displaces only \(10^{-4}\,\text{m}^3\), which is \(7 \times 10^{-4}\,\text{m}^3\) less. Over the floor of \(0.06\,\text{m}^2\), the level falls by \(0.0117\,\text{m}\), from \(15.00\,\text{cm}\) to \(13.83\,\text{cm}\).
A boat carrying a heavy stone floats in a swimming pool. The stone is thrown overboard and sinks to the bottom. What happens to the water level in the pool?
In that pool, the stone has a mass of \(5\,\text{kg}\) and a density of \(2500\,\text{kg/m}^3\). By how many litres does the volume of water displaced fall when it goes overboard?
Buoyancy in a lift
Recall from the chapter on Pressure in Fluids that in a lift accelerating upward at \(a\), the pressure at depth \(h\) is \(P = P_0 + \rho(g + a)h\). Every pressure difference in the liquid is larger by the factor \((g + a)/g\). The buoyant force is a pressure difference, so it grows by the same factor. The sum \(g + a\) is called the effective gravity.
Seen from inside the lift, the floating body also feels a pseudo force \(ma\) downward, so it needs a support of \(m(g + a)\). Both sides of the law of flotation now carry \(g + a\), and it cancels. So a body floats exactly as deep in an accelerating lift as in a lift at rest. In free fall \(g + a = 0\). The liquid then has no pressure differences at all, and there is no buoyant force.
In Fig 4.2 at rest, \(W\) and \(F_B\) are both \(6\,\text{N}\). Accelerating upward, the effective gravity is \(15\,\text{m/s}^2\), and both grow to \(9\,\text{N}\). Accelerating downward, it is \(5\,\text{m/s}^2\), and both shrink to \(3\,\text{N}\). The cube stays \(6\,\text{cm}\) deep in every case.
A stone of mass \(5\,\text{kg}\) and volume \(2 \times 10^{-3}\,\text{m}^3\) hangs on a string, fully under water, in a bucket inside a lift. The lift accelerates upward at \(2\,\text{m/s}^2\). Take \(g = 10\,\text{m/s}^2\). Find the tension in the string, in newtons.
The balloon that leans forward. A car speeds up, and the passengers lean back. A helium balloon on a string in the same car leans forward. The air in the car is pushed towards the back, so its pressure grows towards the back. The balloon is pushed from the high pressure to the low, towards the front. The buoyant force always points against the effective gravity.
Buoyancy in air
Archimedes' principle holds in air as well. Air near the ground has a density of about \(1.2\,\text{kg/m}^3\), so each cubic metre a body occupies gives it \(12\,\text{N}\) of buoyancy. For a stone or a person this is tiny beside the weight. A balloon is different: it occupies a large volume while weighing little.
A balloon of volume \(100\,\text{m}^3\) is filled with helium of density \(0.2\,\text{kg/m}^3\). The air has a density of \(1.2\,\text{kg/m}^3\). The fabric and the basket have a mass of \(60\,\text{kg}\). Take \(g = 10\,\text{m/s}^2\). Find the buoyant force, the largest extra load the balloon can lift, and its first acceleration when it is let go with no extra load.
The balloon displaces \(100\,\text{m}^3\) of air, so \(F_B = 100 \times 1.2 \times 10 = 1200\,\text{N}\).
The helium weighs \(100 \times 0.2 \times 10 = 200\,\text{N}\), and the fabric and basket \(600\,\text{N}\).
The spare lift is \(1200 - 200 - 600 = 400\,\text{N}\), enough for an extra \(40\,\text{kg}\). With no extra load, the total mass is \(80\,\text{kg}\), and \(a = 400/80 = 5\,\text{m/s}^2\) upward.
The balloon feels \(1200\,\text{N}\) of buoyancy, can lift another \(40\,\text{kg}\), and starts upward at \(5\,\text{m/s}^2\). Air drag soon limits its speed.
A toy balloon filled with helium is let go in a room and rises to the ceiling. Why does it rise?
A sky lantern, the akash kandil, is a paper bag with a small flame at its open mouth. The flame heats the air inside to about \(400\,\text{K}\), and hot air is less dense than cold air, as Thermal Physics will show. Say the lantern holds \(0.15\,\text{m}^3\), with air of \(1.2\,\text{kg/m}^3\) outside and \(0.9\,\text{kg/m}^3\) inside. The lift, buoyancy minus the weight of the hot air, is \(0.15 \times 0.3 \times 10 = 0.45\,\text{N}\). That carries \(45\,\text{g}\), so a lantern of \(30\,\text{g}\) of paper and wire rises.
A kilogram of cotton and a kilogram of iron. Put each on a sensitive scale in air, until the scale reads exactly \(1\,\text{kg}\). The cotton fibres, of density about \(1500\,\text{kg/m}^3\), displace about \(0.67\) litre of air, which weighs \(0.8\,\text{g}\). The iron displaces only \(0.13\) litre, \(0.15\,\text{g}\) of air. So the air holds up more of the cotton, and the cotton must have about \(0.65\,\text{g}\) more mass to read the same. Precise weighing corrects for this buoyancy of the air.
The Montgolfier brothers ran a paper mill in the south of France. In June 1783 they lifted a large bag of cloth lined with paper over a fire of straw and wool, before a crowd. They first thought the smoke itself had a lifting power. The lift came only from hot air being less dense than the air around it. In November 1783 a Montgolfier balloon carried two men over Paris, the first free flight of people.
- Expecting a full glass to overflow when floating ice melts.Floating ice displaces its own weight of water and melts into exactly that much water.
- Counting a sunk load as displacing its weight of water.Only a floated load displaces its weight. A sunk load displaces its volume, which is less.
- Forgetting that the weight also scales with \(g + a\) in an accelerating lift.Both the weight and the buoyant force carry \(g + a\), so a floating body keeps its depth.
- Leaving the gas inside a balloon out of its weight.The helium or the hot air must be lifted too. Only the displaced outside air gives the buoyancy.