Density and pressure
| English | Chinese | Pinyin |
|---|---|---|
| density | 密度 | mìdù |
| pressure | 压强 | yāqiáng |
| upthrust | 浮力 | fú lì |
| atmospheric pressure | 大气压强 | dà qì yā qiáng |
| submerged | 浸没 | jìn mò |
| Archimedes' principle | 阿基米德原理 | ā jī mǐ dé yuán lǐ |
| displaced | 排开 | pái kāi |
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 密度, pressure 压强 and upthrust 浮力.
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 precision aneroid barometer measures atmospheric pressure 大气压强
Pressure with depth
p = ρg·h
Pressure is proportional to depth — the gradient is ρg.
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: $\text{Pa} = \dfrac{\text{N}}{\text{m}^2}$.
An iceberg floats with most of its volume hidden: ice is slightly less dense than water
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 ($\approx 1.0 \times 10^{5}\ \text{Pa}$).
A column of liquid of area $A$: its weight sets the extra pressure at the depth below
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.
Match each term to the definition the examiner marks.
Pressure's definition carries the phrase at right angles. Leaving it out is a lost mark even though the formula is the same.
Where upthrust comes from
- Pressure is greater at the bottom of a submerged 浸没 object than at the top.
- That difference is a net upward force — the upthrust.
Archimedes' principle 阿基米德原理
- Upthrust $= \rho_{\text{fluid}}\, g\, V$, where $V$ is the volume of fluid displaced 排开.
- 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.
Worked example: floating and sinking
- A block of volume $2.0\times10^{-3}\ \text{m}^3$ and mass $1.6\ \text{kg}$ is fully submerged in water ($\rho = 1000\ \text{kg/m}^3$). Find the upthrust and the resultant force on it.
- Upthrust is the weight of the fluid displaced: $U = \rho_{\text{fluid}} V g = 1000 \times 2.0\times10^{-3} \times 9.81 = 19.6\ \text{N}$.
- The block's own weight is $mg = 1.6 \times 9.81 = 15.7\ \text{N}$.
- Resultant: $19.6 - 15.7 = 3.9\ \text{N}$ upwards, so the block rises and will float.
- Note whose density each term uses. The upthrust uses the fluid's density and the displaced volume; the weight uses the block's mass. Substituting the block's weight into the upthrust is the standard error.
Marks that slip away
- Upthrust is the weight of the fluid displaced, $\rho_{\text{fluid}} V g$. It is not the weight of the object.
- A floating body displaces its own weight of fluid; a fully submerged one displaces its own volume. Read which case the question describes.
- Pressure is force per unit area with the force at right angles to the surface. The perpendicular part is marked.
- Hydrostatic pressure $p = \rho g h$ depends on depth, not on the shape or width of the container, and $h$ is measured from the surface downwards.
- Density is mass per unit volume. Check the units: a volume in $\text{cm}^3$ with a mass in kg gives an answer wrong by $10^6$.
A block of volume 2.0e-3 m^3 is fully submerged in water of density 1000 kg/m^3. What is the upthrust on it, in N? (g = 9.81 m/s^2)
U = rho_fluid V g = 1000 x 2.0e-3 x 9.81 = 19.6 N. The block's own mass never enters the upthrust; it only decides whether the block rises or sinks.
Hydrostatic pressure at a point in a liquid depends on which of these? Select all that apply.
p = rho g h has no area or shape in it. A narrow tube and a wide tank give the same pressure at the same depth, which surprises people every year.
Put the method for deciding whether a submerged object rises in order.
Each term uses a different body's property: the fluid's density for the upthrust, the object's mass for the weight.
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)