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# Pressure

## Pressure as force over area

Pressure ($P$ or $p$) is the magnitude of the force exerted on an object per unit area of the object's surface: $$\Torange{\text{Pressure}} = \Torange{P} =\dfrac{\Tblue{\text{Force}}}{\Tred{\text{Area}}} = \dfrac{\Tblue{F}}{\Tred{A}}$$

When walking on top of fresh snow in normal shoes, you sink through the snow's surface. Your shoes have a low surface area so the pressure on the surface of the snow is high.

Skis have a greater surface area than normal shoes and so the pressure you apply to the surface of the snow is lower. This allows you to stay on the surface without sinking in.

The walker sinks into the snow while the skier is able to stay on the surface.

Pressure is a scalar, not a vector quantity. In other words, it has no direction.

## Units of pressure

The SI unit of pressure is the pascal $(\text{Pa})$ which is defined as

$$\Torange{\text{Pa}}= \dfrac{\Tblue{\text{N}}}{\Tred{\text{m}^{2}}} = \dfrac{\Tblue{\text{kg m/s}^{2}}}{\Tred{\text{m}^{2}}} = \dfrac{\Torange{\text{kg}}}{\Torange{\text{m s}^{2}}}$$

It is also common to use other (non-SI) units to express pressure:

• The atmosphere ($1\text{ atm}=101.325 \text{ kPa}$), which is equivalent to the standard atmospheric pressure.

• The bar ($1\text{ bar}=10 ^{5} \text{ Pa}$) which is very close to the standard atmospheric pressure ($1\text{ atm}$).

Example Pressure (estimation)
Pressure in a vacuum $0\text{ Pa}$
Blood pressure of healthy adult at rest $16\text{ kPa}$
Atmospheric pressure $100\text{ kPa}$
Water pressure of a garden hose $300\text{ kPa}$
Pressure of the average human bite $1.10\text{ MPa}$

## Pressure due to a fluid column

A fluid is a gas or a liquid. Fluids always take the shape of their container. They have no fixed shape themselves.

A fluid column just means any fluid in a container which has a given height.

The pressure due to a fluid column comes from the weight ($W$) of the fluid. $\Tred{\rho}$ = density of the fluid, $\Tblue{g}$ = gravitational field strength, $\Tviolet{h}$ = depth (or height) of the fluid column and $\Tgreen{A}$ = area . \begin{align*}\Torange{P} =& \dfrac{W}{\Tgreen{A}} = \dfrac{\Tred{\rho} \times \Tblue{g} \times \Tviolet{h} \times \cancel{\Tgreen{A}}}{\cancel{\Tgreen{A}}} \\ \Torange{P} = & \Tred{\rho} \times \Tblue{g} \times \Tviolet{h} \end{align*}

Divers experience an increase in ear pressure when they dive deeper into the sea because the height of the column increases.

The density of water is $1000 \text{ kg/m}^{3}$ and the gravitational field strength is $10 \text{ N/kg}.$ The pressure at a depth of $100 \um$ under the ocean can be calculated: $$\Torange{P} = \Tred{1000 \text{ kg/m}^{3}} \times \Tblue{10 \text{ N/kg}} \times \Tviolet{ 100 \um} = \Torange{1 \, 000 \, 000 \text{ Pa}}$$

The pressure due to the water in the ocean gets larger as you go deeper.

## Measuring atmospheric pressure with a barometer

The atmospheric pressure is the pressure due to the air in the atmosphere. Atmospheric pressure can be measured with a barometer.

A barometer is made by filling a tube with mercury and then placing the tube upside down insde a bath of mercury.

The atmosphere pushes against the mercury bath which pushes the mercury up the tube. The space between the top of the column of mercury and the top of the tube is a vacuum.

The vertical height of the mercury column gives the atmospheric pressure: \begin{align*}\Tred{\text{atmospheric pressure}} = & \text{density of mercury} \times \Tblue{\text{height of column}} \\& \times \Tgreen{\text{gravitational field strength}} \end{align*}

Note that the width of the tube does not matter in this calculation.

A barometer is used to measure the pressure of the atmosphere.

## Comparing gas pressures using a manometer

A manometer measures the difference in pressure between two supplies of gases.

A manometer consists of a U-shaped tube filled with a liquid such as mercury. Each end of the tube is connected to a supply of gas.

The liquid will always be pushed towards the side with the gas that has the lower pressure.

If the two gases have equal pressure, the level of the liquid will be the same on both sides.

A manometer can be used to measure the pressure difference between the atmosphere and a controlled supply of oxygen.

A barometer is a special kind of manometer. One end is connected to the atmosphere and the other is connected to a vacuum.

Left: gas 1 has a higher pressure than gas 2. Right: gas 1 has a lower pressure than gas 2.

## Calculating pressure difference with a manometer

A manometer is a device used to determine the difference in pressure between two gases.

The difference in pressure between the two gases is equal to $$\Torange{\text{density of liquid}} \times \Tblue{\text{height difference}} \times \Tgreen{\text{gravitational field strength}}$$

If the pressure of one of the gases is already known, the pressure of the other gas can be calculated.

The liquid in the diagram is pushed towards the side that is connected to the atmosphere. The pressure of the hydrogen in the diagram is higher than the atmospheric pressure.

The pressure of hydrogen is therefore equal to the atmospheric pressure plus the pressure difference between the two gases.

A manometer can be used to measure the pressure of a supply of hydrogen.

## Hydraulic press

A hydraulic press is a machine that uses the pressure of a fluid to convert a small force acting on a small area into a large force over a large area.

Hydraulic presses are used in industry to forge metals and lift very heavy objects.

A force $F_{1}$ is applied to one piston pushing the liquid deeper into the container. The liquid then pushes up against the second piston with a force $F_{2}.$

The pressure of a fluid inside a closed container is the same everywhere (we can ignore the small pressure differences due to height unless the container is very tall). This means that the pressure on the two pistons in the diagram is the same.

The pressure on each piston is equal to the force divided by the area so $$\Tred{\frac{F_{1}}{A_{1}}} = \Tblue{\frac{F_{2}}{A_{2}}}$$

If $A_{1}$ is much smaller than $A_{2}$ then a small force $F_{1}$ can be used to create a much larger force $F_{2}$.

A hydraulic press converts a small force into a larger one