Pressure in a liquid. Pascal's law. Communicating vessels

Pascal's law, hydrostatic pressure $p = \rho g h$, why it does not depend on the shape of the vessel, communicating vessels.

Terms

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Formulas

Pressure of a liquid at depth

p = \rho g h

Quantities
Symbol	Name	Unit
$p$	pressure of the liquid	Pa
$\rho$	density of the liquid	kg/m³
$g$	acceleration due to gravity	m/s²
$h$	depth (height of the liquid column)	m

Solved for another quantity:

h = \dfrac{p}{\rho g}

\rho = \dfrac{p}{g h}

Note. Here $g \approx 9.8\ \text{m/s}^2$ , in problems $10\ \text{m/s}^2$ is often used. Example ( $g = 10\ \text{m/s}^2$ ): at a depth of $3\ \text{m}$ in water $p = 1000 \cdot 10 \cdot 3 = 30\,000\ \text{Pa} = 30\ \text{kPa}$ . Convert the depth to metres and the density to $\text{kg/m}^3$ ( $1\ \text{g/cm}^3 = 1000\ \text{kg/m}^3$ ). The formula gives only the pressure of the liquid column, without atmospheric pressure.

Lab

Set the quantities with the sliders. The graphs and the formulas with your numbers update at once.

Communicating vessels

The liquid in vessels of different shapes stands at the same level. The red line marks one depth: the pressure on it is the same everywhere.

Graphs

The line p(h) is drawn for the chosen liquid. Click the graph to set the depth.

p(h)

\rho

1,000 kg/m³

h

1 m

p

10,000 Pa = 10 kPa

Formulas with your numbers

p = \rho g h = 1000\cdot 10\cdot 1 = 10000\ \text{Pa}

How to read the graphs

The $p(h)$ graph is a straight line through the origin: at the surface ( $h = 0$ ) the pressure of the liquid column is zero, and it grows in direct proportion to the depth.
If the depth is made $2$ times greater, the pressure becomes $2$ times greater. The ratio $p / h = \rho g$ is the same at every point of the graph.
The slope of the line is $\rho g$ : the denser the liquid, the steeper the graph. In mercury ( $13\,600\ \text{kg/m}^3$ ) the line is $13.6$ times steeper than in water.
The width and the shape of the vessel do not affect the graph: the pressure at a given depth depends only on the density of the liquid and on the depth.

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Terms

Studied 0 of 3

Formulas

Pressure of a liquid at depth

p = \rho g h

Quantities
Symbol	Name	Unit
$p$	pressure of the liquid	Pa
$\rho$	density of the liquid	kg/m³
$g$	acceleration due to gravity	m/s²
$h$	depth (height of the liquid column)	m

Solved for another quantity:

h = \dfrac{p}{\rho g}

\rho = \dfrac{p}{g h}

Lab

Set the quantities with the sliders. The graphs and the formulas with your numbers update at once.

Communicating vessels

The liquid in vessels of different shapes stands at the same level. The red line marks one depth: the pressure on it is the same everywhere.

Graphs

The line p(h) is drawn for the chosen liquid. Click the graph to set the depth.

p(h)

\rho

1,000 kg/m³

h

1 m

p

10,000 Pa = 10 kPa

Formulas with your numbers

p = \rho g h = 1000\cdot 10\cdot 1 = 10000\ \text{Pa}

How to read the graphs

The

p(h)

graph is a straight line through the origin: at the surface (

h = 0

) the pressure of the liquid column is zero, and it grows in direct proportion to the depth.

If the depth is made

2

times greater, the pressure becomes

2

times greater. The ratio

p / h = \rho g

is the same at every point of the graph.

The slope of the line is

\rho g

: the denser the liquid, the steeper the graph. In mercury (

13\,600\ \text{kg/m}^3

) the line is

13.6

times steeper than in water.

The width and the shape of the vessel do not affect the graph: the pressure at a given depth depends only on the density of the liquid and on the depth.