Pressure and Fluids at Rest: Quick Revision

A brief note on Pressure and Fluids at Rest. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Pressure: $p = \dfrac{F}{A}$, in pascal. A fluid pushes at right angles to every surface.
• Depth: $p = p_0 + \rho gh$, the same at every point at one depth, whatever the vessel's shape.
• Pascal's law: a change of pressure reaches every part of an enclosed fluid, so $F_2 = F_1\dfrac{A_2}{A_1}$.
• The air: $p_0 = 1.013 \times 10^{5}\ \text{Pa}$, $76\ \text{cm}$ of mercury. Gauge pressure is the part above it.
• Archimedes: the buoyant force is the weight of fluid pushed aside; a floating body has $\dfrac{\rho_b}{\rho_f}$ of its volume under.

The facts at a glance

A worked example

A wooden buoy of volume $0.080\ \text{m}^3$ and density $450\ \text{kg/m}^3$ floats in the sea ($1030\ \text{kg/m}^3$).

1. Mass. $450\ \text{kg/m}^3 \times 0.080\ \text{m}^3 = 36\ \text{kg}$.
2. Fraction under. $\dfrac{450\ \text{kg/m}^3}{1030\ \text{kg/m}^3} = 0.437$, so $43.7\%$ is under.
3. Extra load. Fully under it pushes aside $1030\ \text{kg/m}^3 \times 0.080\ \text{m}^3 = 82.4\ \text{kg}$ of sea, so it can carry $82.4\ \text{kg} - 36\ \text{kg} = 46.4\ \text{kg}$ before it goes under.

Practice with new numbers

A program to try

The program works out the buoy of the worked example.

Mistakes to avoid

• Gauge for absolute pressure, or the other way round. Read the question for which one it wants.
• Diameters for areas in a hydraulic lift. Square the ratio.
• The vessel's shape changing the pressure. Only the depth matters.
• The body's density in the buoyant force. Use the fluid's.
• Forgetting the air behind a window or a mask. It cancels the air outside.

Check yourself

1. What is the gauge pressure $7.0\ \text{m}$ down in fresh water?

2. A lift's pistons are $3\ \text{cm}$ and $30\ \text{cm}$ across. What force on the small one holds up $5000\ \text{N}$?

3. A barometer reads $72\ \text{cm}$ of mercury. Is the place above or below sea level?

4. A block floats in water with $\tfrac13$ of its volume above the surface. What is its density?

5. Why does a helium balloon rise?