At the container terminal of Mumbai port, a ship made of thousands of tonnes of steel floats with stacks of containers on its deck. A steel nail dropped over its side goes straight to the bottom of the harbour. Both are steel, and steel is nearly eight times as dense as water.
Floating is a balance
Let go a body less dense than water with its bottom just touching the surface. At first almost none of it is under water, and the buoyant force is tiny. So the body sinks. As more of it goes under, it displaces more water, and the buoyant force grows. The body stops sinking when the buoyant force has grown to equal its weight. This state of balance is called flotation.
The cube weighs \(600 \times 0.001 \times 10 = 6\,\text{N}\). Its base has an area of \(0.01\,\text{m}^2\), so each centimetre under water displaces \(10^{-4}\,\text{m}^3\) of water, which weighs \(1\,\text{N}\). The buoyant force therefore reaches \(6\,\text{N}\) at \(d = 6\,\text{cm}\), where the line of the graph crosses \(W\). The cube arrives there moving, so it overshoots to about \(8.4\,\text{cm}\) and bobs. Water drag makes the bobbing die away, and it settles at \(6\,\text{cm}\).
A body floating at rest in a fluid displaces a weight of fluid equal to its own weight.
A boat of mass \(200\,\text{kg}\) floats on a lake. (a) What volume of water does it displace? (b) Cargo of \(100\,\text{kg}\) is put in it. What volume does it displace now? (c) The boat has vertical sides near the waterline, and its waterline area is \(2\,\text{m}^2\). By how much does it sink when the cargo is loaded?
(a) The boat displaces its own weight of water, \(2000\,\text{N}\), so \(V = 200/1000 = 0.2\,\text{m}^3\).
(b) The boat and cargo have a mass of \(300\,\text{kg}\), so they displace \(300/1000 = 0.3\,\text{m}^3\).
(c) The extra \(0.1\,\text{m}^3\) is a thin slab over an area of \(2\,\text{m}^2\), so the boat sinks \(0.1/2 = 0.05\,\text{m} = 5\,\text{cm}\).
A ship sails from a river into the sea, where the water is denser. What happens to the buoyant force on it?
How much is under
Divide both sides of the law of flotation by \(\rho_f V g\). The fraction of the body under the surface is the ratio of the two densities. Wood of \(600\,\text{kg/m}^3\) floats with \(0.6\) of its volume under water. In oil of \(800\,\text{kg/m}^3\) it floats with \(600/800 = 0.75\) under. In a lighter liquid more of the body must go under to displace the same weight.
Set the slider in the sim above to \(900\,\text{kg/m}^3\), a round value for ice. In fresh water \(0.90\) of the block is under. In sea water only \(900/1025 = 0.88\) is under, so a little more shows. This is the hidden bulk of an iceberg: almost nine tenths of it lies below the sea. Now set \(1000\) in fresh water. The block hangs fully under, wherever it is put. Above \(1000\) it goes to the bottom.
A body floats with \(60\,\%\) of its volume under water, and with \(75\,\%\) of its volume under the surface of another liquid. Find the density of the other liquid, in kg/m³.
A cylinder \(20\,\text{cm}\) tall floats upright in water with \(16\,\text{cm}\) of its height under. It is moved to oil of density \(800\,\text{kg/m}^3\). How does it float?
Pushing a float under
A cube of side \(10\,\text{cm}\) and density \(500\,\text{kg/m}^3\) floats with half of it, \(5\,\text{cm}\), under water. Push it down until it is fully under. Now it displaces \(10^{-3}\,\text{m}^3\) of water, and the buoyant force is \(10\,\text{N}\). Its weight is only \(5\,\text{N}\), so your hand must push down with the difference, \(5\,\text{N}\). That is the weight of the extra water the cube now displaces.
A wooden block of volume \(2 \times 10^{-3}\,\text{m}^3\) and density \(600\,\text{kg/m}^3\) floats in water. Take \(g = 10\,\text{m/s}^2\). What downward force holds it just fully under the surface, in newtons?
River water and sea water
Sea water, at about \(1025\,\text{kg/m}^3\), is denser than river water. So a ship floats higher in the sea. A ship loaded at Kolkata port, on the Hooghly, rises as it reaches the salt water of the Bay of Bengal. Every cargo ship carries a load line painted on its hull. Its marks show how deep the ship may lie in fresh water and in sea water.
A loaded ship of mass \(10\,000\,\text{t}\) floats at Kolkata port in river water of \(1000\,\text{kg/m}^3\). Its waterline area is \(2000\,\text{m}^2\), and its sides are vertical near the waterline. By how much does it rise when it reaches sea water of \(1025\,\text{kg/m}^3\), in cm?
A bowl of steel
Steel is \(7.8\) times as dense as water, so a solid lump of steel sinks. Press the same steel into a bowl, and it encloses a large volume of air. What decides floating is the average density of the body: its total mass divided by the whole volume it encloses, the air included. A body floats when its average density is less than the density of the liquid.
The block in Fig 2.2 can push aside at most \(30\,\text{cm}^3\) of water, which weighs \(30\,\text{g}\), against its own \(234\,\text{g}\). So it sinks. The bowl encloses \(10 \times 10 \times 5 = 500\,\text{cm}^3\), and its average density is \(234/500 = 0.47\,\text{g/cm}^3\), less than that of water. It floats with \(234/100 = 2.34\,\text{cm}\) of its depth under water, where it displaces \(234\,\text{g}\) of water. A steel ship is this bowl made enormous.
A hollow steel ball has a mass of \(2\,\text{kg}\) and an outer volume of \(4 \times 10^{-3}\,\text{m}^3\). It floats in water. What fraction of its volume is under the surface?
Return to the ship at Mumbai port. Its steel hull encloses a huge volume of air, so the average density of hull and cargo together stays below that of sea water. It sinks until it has displaced its own weight, and then it floats. The nail has no air inside. Its average density is that of steel, and no depth of water can hold it up.
A loaded container ship at Mumbai port has a mass of about \(50\,000\,\text{t}\). Taking sea water as \(1025\,\text{kg/m}^3\), estimate the volume of sea water it displaces, in cubic metres.
In the 1860s many British cargo ships sailed so overloaded that they sank in the first storm. Some owners insured such ships for more than their worth, and the sailors paid with their lives. Samuel Plimsoll, a member of Parliament, fought a long campaign against these coffin ships. In 1876 a law required British ships to carry a load line on the hull. The mark still bears his name.
- Using the whole volume of a floating body in \(F_B = \rho_f V g\).Only the part under the surface displaces fluid. Use \(V_{sub}\).
- Thinking a ship rides higher in the sea because the buoyant force there is larger.The buoyant force equals the weight in both. The denser sea supplies it from a smaller volume.
- Deciding whether a steel ship floats from the density of steel.Use the average density, the total mass over the whole volume the hull encloses.