Floating objects, ships and balloons

A steel nail sinks, yet a steel ship stays afloat. Archimedes' principle is the same. How does the comparison of forces change when the same material forms a large hollow hull?

Hold an object inside a liquid, then release it from rest. Gravity mg pulls down; buoyant force FА acts up. If FА > mg, the unbalanced force points up and the object starts to rise; if FА < mg, it starts to sink. With equal forces, it can remain at rest. We include only these two forces here: a string, support, significant current or drag would require us to add forces to the diagram.

Cut a solid steel cube, then make a sealed box from the same mass of steel. The mass is nearly unchanged but the external volume is larger, so the box can displace more water. Average density ρт = m/Vвнешний compares total mass with external volume, including sealed cavities; its unit is kg/m³. A ship's mass includes hull, air, equipment and cargo. Steel density alone does not describe the entire ship.

At full immersion, FА = ρжgV and mg = ρтVg. Both expressions contain the same g and external displaced volume V, so comparing forces becomes comparing densities. If ρт < ρж, the released object starts to rise; if ρт > ρж, it starts to sink. Equal densities allow equilibrium inside the liquid. This assumes that the chosen volume really displaces liquid; open cavities that fill with liquid need separate treatment.

If a light object rises, why does it not fly out of the water? As it emerges through the surface, its immersed volume decreases. FА then decreases until FА = Fтяж. This is the condition for floating at rest. Greater immersion increases support; shallower immersion reduces it. After possible oscillations have died away, the object settles at equilibrium. A momentary equality of forces during motion does not by itself mean the object has stopped.

Put cargo into a boat floating at rest. Its gravitational force increases, so the previous buoyant force is insufficient: the boat sinks deeper, displacing more water until the forces balance again. A hollow hull provides a large displaced volume for a moderate total mass. But the available extra immersion is limited. If water starts entering, the calculation for a dry hull no longer applies: both the system's mass and the volume available to keep water out change.

In air, buoyancy must support everything that rises: the gas inside, envelope, basket and load. To start rising, FА must exceed their combined gravitational force. A light gas reduces mass while keeping a large volume. In a hot-air balloon, heated air expands and some escapes through the open bottom, reducing the mass of air inside the envelope at a given volume. Heating a sealed vessel does not by itself reduce mass; lift depends on the balloon's construction, not simply on being 'hot'.

Throw a light object down into water. If buoyancy exceeds gravity, must it immediately move upwards? At first it may continue downwards while slowing, because force changes velocity rather than resetting it at every instant. Its motion may then reverse. That is why the simple rising and sinking rules above refer to release from rest. For an object already moving, comparing forces tells us the direction in which velocity changes.

A block of average density 600 kg/m³ floats at rest in water of density 1000 kg/m³. Find the fraction of its volume that is immersed.

For floating equilibrium, ρводы·g·Vпогр = m·g. The block's mass is ρт·Vвесь.

Cancelling g gives Vпогр/Vвесь = ρт/ρводы = 600/1000 = 0,6.

Six tenths of the volume displaces water whose weight equals the block's gravitational force.

60% of the volume is underwater. This part displaces water with the same mass as the entire block; the remaining 40% of the volume is above the surface.

Floating depends on comparing the entire system's gravitational force with buoyancy, not just naming its material.

For floating at rest, FА = mg; adding cargo requires more water to be displaced.

Forces determine changes in velocity. A rule for starting to rise from rest does not instantly specify the direction of existing motion.

Forces transfer energy when lifting loads. The next lesson introduces mechanical work and power.