Buoyancy

Buoyancy is the upward force a fluid (liquid or gas) exerts on any object that is fully or partially immersed in it. The force itself is called the buoyant force, or upthrust.

Why an upward force appears at all. As covered in "Pressure in fluids," the pressure at a point inside a fluid grows with depth — a deeper point has more fluid weighing down on it. Now picture a solid block lowered into water. Its bottom face sits at a greater depth than its top face, so the fluid pressure pushing up on the bottom is larger than the fluid pressure pushing down on the top. Pressure acting over area gives force, so this depth-driven pressure gap becomes a net upward force on the block — the buoyant force. Sideways pressures on the block's vertical faces are equal and opposite at every depth, so they cancel; only the top-versus-bottom imbalance survives.

where: $P_{\text{bottom}}$ and $P_{\text{top}}$ are the fluid pressures on the object's bottom and top surfaces, $F_{\text{up}}$ and $F_{\text{down}}$ are the resulting forces on those surfaces, and $F_B$ is the buoyant force — the net upward force left over once the downward push is subtracted from the upward push.

Archimedes' Principle packages this reasoning into one usable statement (kept qualitative at Class 9 level): when a body is immersed, wholly or partially, in a fluid, it experiences an upward buoyant force equal to the weight of the fluid displaced by the body. In plain words: push an object into water and it shoves some water out of the way — it displaces that water. Archimedes' Principle says the buoyant force on the object exactly equals how much that displaced water would weigh, no more and no less.

This single principle explains a wide range of everyday observations: why objects feel lighter when submerged (the buoyant force partly cancels their weight); why a steel ship still floats despite being denser than water, provided its hollow shape displaces enough water to match the ship's own weight; and how a hydrometer — a sealed, weighted tube that floats upright in a liquid — floats higher in a denser liquid, since less volume needs to be pushed aside to match its weight. Whether an object floats, sinks, or hangs suspended depends on comparing its own density with the fluid's density — the subject of the next article, "Floating and sinking of objects."

Here is the reasoning laid out step by step, connecting the depth-pressure fact to the full statement of Archimedes' Principle, entirely without needing a numerical density formula.

Step 1. Start from the established fact that fluid pressure increases with depth. In a fluid at rest, every point at a greater depth carries a larger pressure than a point closer to the surface, because there is more fluid weight stacked above it.

Step 2. Apply this to an immersed object. Whatever the object's shape, its lowest points sit deeper than its highest points. The fluid therefore pushes up on the lower surfaces with more pressure than it pushes down on the upper surfaces.

Step 3. Convert the pressure imbalance into a force imbalance. Pressure multiplied by the area it acts on gives a force; since the upward-pushing pressures are systematically larger than the downward-pushing ones, the forces do not cancel — a net force survives, and it points upward. This is the buoyant force, and it exists purely because the object occupies space at different depths in the fluid, regardless of what the object is made of.

Step 4. Recognise what fixes the size of this net force: it is exactly what is needed to support the weight of the fluid that used to occupy the space the object now takes up — the displaced fluid. This is the content of Archimedes' Principle, and it is why the buoyant force depends on how much fluid the object pushes aside (its immersed volume), not on how deep it happens to be sitting or what material it is made of.

Step 5. This also explains why a fully submerged object's buoyant force stays constant as it is pushed deeper: it keeps displacing the same volume of fluid at every depth, so the weight of displaced fluid stays constant even though the absolute pressures on its top and bottom both keep rising together.

A stone is lowered into a bucket that is full of water right up to the brim, and exactly $300\ \text{mL}$ of water overflows as the stone goes in. A hydrometer is then tested first in this same fresh water and afterwards in salty seawater, which is denser than fresh water. Explain, using Archimedes' Principle, (a) the buoyant force on the stone, and (b) which liquid the hydrometer floats higher in.

Step 1. Identify the displaced volume for the stone. The $300\ \text{mL}$ of water that overflowed is exactly the volume of water the stone pushed out of the way as it went in — the displaced volume.

Step 2. Apply Archimedes' Principle to the stone. The buoyant force on the stone equals the weight of this displaced water. Since $1\ \text{mL}$ of water has a mass of about $1\ \text{g}$, the displaced water has a mass of about $300\ \text{g}$, so the buoyant force is the weight of roughly $300\ \text{g}$ of water acting upward on the stone — this is why the stone would feel noticeably lighter lifted underwater than lifted in air. (Class 9 stops at "the weight of $300\ \text{g}$ of water"; converting this to newtons using $F_B = \rho_{\text{liquid}} V g$ is Class 10 treatment.)

Step 3. Reason about the hydrometer using the same principle. It floats once the buoyant force — the weight of liquid it displaces — equals its own fixed weight. Seawater is denser than fresh water, so each millilitre of seawater weighs more than a millilitre of fresh water.

Step 4. Compare the two liquids. Because each millilitre of seawater is heavier, the hydrometer needs to displace a smaller volume of seawater to reach a displaced weight equal to its own weight. A smaller displaced volume means less of the hydrometer needs to sink in, so it rides higher in seawater than in fresh water.

The buoyant force on the stone is the weight of about 300 g of displaced water, and the hydrometer floats higher in the denser seawater because it needs to push aside less liquid to balance its own weight.

• Thinking buoyant force depends on depth. Once an object is fully submerged, pushing it deeper does not change the buoyant force — it depends on the displaced volume, which stays the same, not on how deep that volume happens to be.
• Confusing "displaced fluid" with "fluid that spills out." Displacement just means the fluid that would have occupied the object's immersed volume; it does not need to physically overflow out of an open container to count, though it conveniently does in a brim-full bucket.
• Assuming only floating objects feel a buoyant force. Every immersed object experiences an upward buoyant force, sinking ones included — whether it floats or sinks depends on comparing this force with the object's own weight, not on whether a buoyant force exists at all.
• Reaching for a density-times-volume-times-g numerical formula. At Class 9 level, Archimedes' Principle is used and explained in words and comparisons, not computed numerically with $F_B = \rho_{\text{liquid}} V g$; that calculation belongs to Class 10.

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