Density and pressure
Why a steel ship floats
- A solid steel nail sinks, yet a huge steel ship floats.
- The ship's shape pushes aside a lot of water — and the water pushes back.
- To explain it we need density, pressure and upthrust.
Density
- Density is mass per unit volume: $\rho = \dfrac{m}{V}$, unit $\dfrac{\text{kg}}{\text{m}^3}$.
- Worth knowing: water $1000$, air $\approx 1.2$, steel $\approx 7800\ \dfrac{\text{kg}}{\text{m}^3}$.
A block has mass $5.0\ \text{kg}$ and volume $0.0020\ \text{m}^3$. What is its density?
$\rho = \dfrac{m}{V} = \dfrac{5.0}{0.0020} = 2500\ \dfrac{\text{kg}}{\text{m}^3}$.
Pressure
- Pressure is force per unit area, at right angles to the surface: $p = \dfrac{F}{A}$.
- Unit: $\text{Pa} = \dfrac{\text{N}}{\text{m}^2}$.
Pressure is the force per unit ____ (acting at right angles to the surface).
$p = \dfrac{F}{A}$, measured in $\text{Pa} = \dfrac{\text{N}}{\text{m}^2}$.
Pressure in a liquid
- The extra pressure at a depth is $\Delta p = \rho g \Delta h$.
- It depends only on the density and depth — not the container's shape.
- For the total pressure, add the atmospheric pressure ($\approx 1.0 \times 10^{5}\ \text{Pa}$).
The extra pressure at a depth in a liquid depends on:
$\Delta p = \rho g \Delta h$ — only density and depth matter; the container shape does not.
Where upthrust comes from
- Pressure is greater at the bottom of a submerged object than at the top.
- That difference is a net upward force — the upthrust.
Archimedes' principle
- Upthrust $= \rho_{\text{fluid}}\, g\, V$, where $V$ is the volume of fluid displaced.
- In words: the upthrust equals the weight of the fluid pushed aside.
- An object floats when this upthrust equals its weight.
Archimedes' principle: the upthrust equals the weight of the fluid that is ____.
The upthrust $= \rho g V$, where $V$ is the displaced volume — the weight of fluid pushed out of the way.
A block of volume $0.0020\ \text{m}^3$ is fully under water ($\rho = 1000\ \dfrac{\text{kg}}{\text{m}^3}$, $g = 9.81\ \dfrac{\text{m}}{\text{s}^2}$). What is the upthrust?
Upthrust $= \rho g V = 1000 \times 9.81 \times 0.0020 \approx 19.6\ \text{N}$.
Weighing in a fluid
- A submerged block on a newton-meter reads less than its real weight: reading $= W - \text{upthrust}$.
- That is why heavy things feel lighter under water.
A floating object displaces its own weight of fluid.
Yes — it sinks until the upthrust (weight of fluid displaced) equals its own weight, then floats.
You've got it
- density $\rho = \dfrac{m}{V}$; pressure $p = \dfrac{F}{A}$
- pressure in a liquid: $\Delta p = \rho g \Delta h$ — set by density and depth only
- upthrust $= \rho g V$ = the weight of fluid displaced (Archimedes)