SPI Physics · Fluid Mechanics and Properties of Matter

Thrust and Liquid Pressure

Pressure is the force pressing on each square metre of a surface, and inside a liquid at rest it depends only on how deep you are and how dense the liquid is.

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Why this matters

Three questions in the bank, one EASY and two MODERATE: which faces of an iron cuboid press equally on sand, two points at the same depth in a tank, and the pressure under two layers of liquid. All three come down to P equals F over A and P equals hρg, used with care over units.

Concept 1 of 2: Thrust, pressure and the pressure of a liquid column

A sharp knife cuts and a blunt one does not, though you push just as hard. The same force on a smaller area presses harder. Pressure measures exactly that: force shared out over area.

Definition

Thrust is the force acting at right angles to a surface. Its unit is the newton (N).
Pressure $=$ thrust $\div$ area. Its unit is the pascal: $1\ \text{Pa} = 1\ \text{N/m}^2$ .
The same weight on a smaller face gives a larger pressure. Opposite faces of a cuboid have equal areas, so they give equal pressures.
A column of liquid of height h presses on its base with its weight. That gives a pressure of $h\rho g$ , where $\rho$ is the density of the liquid.
With two layers of liquid, work out $h\rho g$ for each layer and add them.
Change a density in g/cm³ (g/cc) to kg/m³ by multiplying by 1000: water is 1 g/cm³ $= 1000\ \text{kg/m}^3$ .

Pressure

P = \frac{F}{A} \qquad P_{\text{liquid}} = h\rho g \qquad 1\ \text{g/cm}^3 = 1000\ \text{kg/m}^3

Fthrust (force at right angles), in newtons
Aarea, in square metres
hdepth of liquid, in metres
ρdensity of the liquid, in kg/m³

Worked example

A block weighing 120 N measures 0.2 m by 0.3 m by 0.5 m. Find the greatest and the least pressure it can put on the floor.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Fluid Mechanics and Properties of Matter · Thrust and Liquid Pressure

A tank 2 m high is half filled with water and then filled to the top with oil of density 0.60 g/cc. What is the pressure at the bottom of the tank due to these liquids? (Take g =

10\ \text{m s}^{-2}

)

Using g/cm³ in h ρ g

With h in metres and g in m/s², the density must be in kg/m³. An oil of 0.9 g/cm³ is 900 kg/m³; leaving it as 0.9 makes the answer a thousand times too small.

Thinking a heavier-looking face presses harder

The weight is the same whichever face is down. Only the area changes, so the smallest face gives the largest pressure, and opposite faces give the same pressure.

Drill 1 more on thrust, pressure and the pressure of a liquid column

Concept 2 of 2: Pressure in a liquid depends only on depth

A diver feels the same squeeze in the middle of a pool and near its wall, as long as she is at the same depth. What matters is how much liquid is stacked above her, not how wide the pool is or where she is in it.

Definition

Pressure in a liquid at rest increases with depth.
At the same depth in the same liquid, the pressure is the same everywhere: near the wall or in the middle, in a wide jar or a narrow one.
At any point it acts equally in all directions, sideways and upwards as well as down.
In a denser liquid, the pressure at the same depth is larger.
This is why a dam wall is made thicker at the bottom.

Pressure at a depth

P = P_0 + h\rho g

P₀air pressure on the open surface
hdepth below the free surface

Worked example

A wide jar and a narrow jar are both filled with water to a height of 0.3 m. Compare the pressure due to the water at their bottoms. Take

\rho = 1000\ \text{kg/m}^3

and

g = 10\ \text{m/s}^2

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Fluid Mechanics and Properties of Matter · Thrust and Liquid Pressure

In a tank having 'h' height of liquid, two points at a depth

\frac{3h}{4}

from free surface have pressure

P_A

and

P_B

. When point A is closer to the wall than B. Then

Thinking the wall changes the pressure

Two points at the same depth have the same pressure, whether one is near the wall and the other in the middle. Only depth and the liquid's density matter.

Thinking a wider vessel gives more pressure

A wide tank holds more water than a narrow one of the same height, but the pressure at the bottom is the same. The extra water is spread over a larger base.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Thrust, pressure and the pressure of a liquid column
Pressure
$P = \frac{F}{A} \qquad P_{\text{liquid}} = h\rho g \qquad 1\ \text{g/cm}^3 = 1000\ \text{kg/m}^3$
Pressure in a liquid depends only on depth
Pressure at a depth
$P = P_0 + h\rho g$

Watch out for (4)

Using g/cm³ in h ρ g
→ Thrust, pressure and the pressure of a liquid column
Thinking a heavier-looking face presses harder
→ Thrust, pressure and the pressure of a liquid column
Thinking the wall changes the pressure
→ Pressure in a liquid depends only on depth
Thinking a wider vessel gives more pressure
→ Pressure in a liquid depends only on depth

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