2.1 - Pressure in Liquids

Liquid Pressure and Its Derivation

• Liquid pressure arises from the weight of the liquid column above a point.
• For a column of height $h$, base area $A$ and density $\rho$:

Symbol	Meaning and SI unit
$P$	Liquid pressure, $\mathrm{Pa}$
$h$	Depth below the liquid surface, $\mathrm{m}$
$\rho$	Liquid density, $\mathrm{kg}{\mathrm{m}}^{-3}$
$g$	Gravitational acceleration, $\mathrm{m}{\mathrm{s}}^{-2}$

• Pascal is the SI unit of pressure: $1\mathrm{Pa}=1\mathrm{N}{\mathrm{m}}^{-2}=1\mathrm{kg}{\mathrm{m}}^{-1}{\mathrm{s}}^{-2}$.

Factors Affecting Liquid Pressure

• At fixed $\rho$ and $g$, greater depth produces greater pressure.
• At fixed $h$ and $g$, greater density produces greater pressure.
• Pressure also depends on $g$.

Experiment 2.1: Depth and Pressure

• Aim: Study the relationship between liquid depth and pressure.
• Inference: Liquid pressure depends on depth.
• Hypothesis: Greater depth gives greater liquid pressure.
• Variables: Manipulated — depth $h$; responding — pressure represented by the difference $D$ between the two water levels in a U-tube; constant — liquid density.
• Apparatus: 500 ml measuring cylinder, silicone tube, thistle funnel covered with a thin rubber sheet, U-tube, two half-metre rules and retort stand.
• Materials: Water and food colouring.

1. Connect the covered thistle funnel to the U-tube using the silicone tube. Initially keep the funnel outside the measuring cylinder, with equal water levels in the U-tube.
2. Immerse the funnel mouth to $h=4.0\mathrm{cm}$ and measure $D$.
3. Repeat at $8.0$, $12.0$, $16.0$ and $20.0\mathrm{cm}$.
4. Record $h$ and $D$, then plot $D$ against $h$.

Experiment 2.2: Density and Pressure

• Aim: Study the relationship between liquid density and pressure.
• Inference: Liquid pressure depends on density.
• Hypothesis: Higher density gives higher pressure.
• Variables: Manipulated — density $\rho$; responding — pressure represented by the U-tube water-level difference $D$; constant — liquid depth.
• Apparatus: Half-metre rule, three 600 ml beakers, U-tube, silicone tube, thistle funnel covered with a thin rubber sheet and retort stand.
• Materials: Masking tape, alcohol, water and glycerine.

1. Mark each beaker $2\mathrm{cm}$ above its base. Use beakers A, B and C for alcohol, water and glycerine respectively.
2. Connect the covered funnel to the water-filled U-tube. Slowly immerse the funnel vertically in alcohol until its mouth is level with the mark.
3. Measure $D$, then remove and dry the funnel.
4. Repeat for water and glycerine. Record $D$ for each liquid and relate it to density.

Liquid	Density / $\mathrm{kg}{\mathrm{m}}^{-3}$
Alcohol	790
Water	1 000
Glycerine	1 300

Characteristics of Liquid Pressure

• Liquid pressure does not depend independently on the liquid’s mass, volume, surface area or the shape of its container.
• Pressure at a point acts in all directions. Points at the same level in the same static liquid have equal pressure.
• Connected columns of different shapes and cross-sectional areas have the same liquid level. Holes at the same level in a water-filled bottle produce equal horizontal spurt distances.
• For two immiscible liquids in a U-tube, compare pressures at the same horizontal level in the connected static liquid. With both free surfaces exposed to the same atmospheric pressure:

• Measure each column height from its free surface to the common reference level.

Actual Pressure and Worked Examples

• A liquid exposed to the atmosphere experiences pressure at its surface. Actual pressure at depth includes both contributions:

• A fish at $1.5\mathrm{m}$ in aquarium water of density $1050\mathrm{kg}{\mathrm{m}}^{-3}$, with $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$ and $P_{\mathrm{atm}}=100\mathrm{kPa}$:

• In a U-tube, a $15.0\mathrm{cm}$ water column balances a $16.5\mathrm{cm}$ olive-oil column. With ${\rho }_{\text{water}}=1000\mathrm{kg}{\mathrm{m}}^{-3}$:

Applications

• Household water tank: Placing it between the ceiling and roof gives a large height difference above the tap, producing pressure that makes water flow quickly.
• Intravenous liquid: A bag above the patient’s body produces pressure that pushes liquid into the body. Bag height affects flow rate; an adjuster controls the flow.
• Dam: A thicker base withstands the greater pressure at greater depth. A low penstock supplies fast-flowing water to drive turbines.
• Siphon: A water-filled flexible tube transfers water from a higher reservoir to a lower outlet without electrical power. Outflow lowers pressure near the top of the tube; atmospheric pressure pushes water into its submerged inlet.
• Rate of liquid transfer is volume transferred per unit time:

• With $V$ in ml and $t$ in seconds, the unit is $\mathrm{ml}{\mathrm{s}}^{-1}$.