Floating and Sinking

Since pressure determines force ($F = PA$), this creates a larger upward force on the bottom than the downward force on the top. This difference results in a net upward force. This principle applies to all fluids, which is why a helium balloon experiences upthrust in the air just as a submarine does in the ocean.

Archimedes' Principle and Displacement

When you lower an object into a fluid, it pushes fluid out of the way to make room for itself. This process is called displacement. According to Archimedes' Principle, the upthrust acting on an object is exactly equal to the displaced fluid weight.

As more of an object is submerged, it displaces a greater volume of fluid, which directly increases the upthrust force. An object reaches its maximum possible upthrust when it is fully submerged. A floating object only displaces a volume of fluid equal to its submerged portion, whereas a completely submerged object displaces a volume equal to its entire size.

The Conditions for Floating

An object will float when its weight is perfectly balanced by the upthrust acting upon it ($W = U$). In this state, the forces are balanced, so the resultant force is zero and the object remains stationary at the surface.

For an object to float, its density must be less than the density of the fluid. Large ships made from dense steel can float because they contain vast air-filled spaces. This reduces their average density to less than the density of water (approximately $1000\text{ kg/m}^3$).

If an object is placed in a denser fluid, like seawater, it needs to displace a smaller volume of fluid to match its own weight. Therefore, objects float higher in denser fluids.

The Conditions for Sinking

If an object is denser than the fluid it is placed in, it cannot displace a weight of fluid equal to its own weight. As a result, its weight is greater than the maximum possible upthrust.

This imbalance creates a downward resultant force (Weight $-$ Upthrust), causing the object to accelerate downwards and sink. Crucially, sinking objects do not have balanced forces, and the upthrust they experience will never equal their total weight.

Worked Example: Floating and Sinking

A metal cylinder has a volume of $0.05\text{ m}^3$ and a weight of $600\text{ N}$. It is fully submerged in water, which has a density of $1000\text{ kg/m}^3$. Calculate the maximum upthrust acting on the cylinder and explain whether it will float or sink. Assume gravitational field strength is $10\text{ N/kg}$.

Step 1: State the formula for the maximum upthrust (Higher Tier).

$$U = V_s \times \rho_f \times g$$

Step 2: Identify the values and substitute them into the equation.

• $V_s = 0.05\text{ m}^3$, $\rho_f = 1000\text{ kg/m}^3$, $g = 10\text{ N/kg}$
• $U = 0.05 \times 1000 \times 10$

Step 3: Calculate the maximum upthrust.

• $U = 500\text{ N}$

Step 4: Compare the upthrust to the weight to determine the outcome.

• The downward weight ($600\text{ N}$) is greater than the maximum upward upthrust ($500\text{ N}$).
• There is a downward resultant force of $100\text{ N}$, meaning the cylinder will sink.