Teaching guide for the module

Fluid mechanics: Pascal, Boyle-Mariotte, Archimedes: Teacher's Guide

Teaching guide for the Fluid Dynamics simulator: explain in class Pascal's Law and hydraulic systems, the Boyle-Mariotte Law on ideal gases at constant temperature, and Archimedes' Principle on the buoyancy of bodies in fluids. Designed for physics and mechanical technology teachers.

Module: Fluid Dynamics · Three tabs: Pascal's Law · Boyle-Mariotte · Archimedes

Physical phenomenon

Fluid mechanics (liquids and gases) rests on three cornerstone laws, each one the subject of a tab in the module.

Pascal's Law. An incompressible liquid in equilibrium transmits unchanged the pressure applied at any of its points to all other points of the fluid. Fundamental consequence: in a hydraulic system with two pistons of different areas $A_{1}$ and $A_{2}$ connected,

$\frac{F _{1}}{A _{1}} = \frac{F _{2}}{A _{2}} ⟹ F_{2} = F_{1} \cdot \frac{A _{2}}{A _{1}}$

This is the principle of the hydraulic force multiplier: the basis of all hydraulics.

Boyle-Mariotte Law. For an ideal gas at constant temperature, the product of pressure and volume is constant:

$P \cdot V = const$

The $P$ - $V$ graph is an equilateral hyperbola. Compressing the gas to half its initial volume doubles the pressure.

Archimedes' Principle. A body immersed in a fluid feels an upward buoyancy force equal to the weight of the displaced fluid:

$F_{A} = ρ_{fluid} \cdot g \cdot V_{immersed}$

Comparing $F_{A}$ with the weight $P = ρ_{body} \cdot g \cdot V$ , you determine whether the body floats ( $ρ_{body} < ρ_{fluid}$ ), sinks ( $ρ_{body} > ρ_{fluid}$ ), or stays suspended ( $ρ_{body} = ρ_{fluid}$ ).

Key concepts

Pressure: ratio of force to surface area, $P = F / A$ . Measured in pascal (Pa).
Hydraulic mechanical advantage: ratio $A_{2} / A_{1}$ between piston areas; can be very large without complex mechanical devices.
Volume conservation: in a closed hydraulic system, the liquid displaced by the small piston fills the large one: "long" displacement turns into "short" displacement with multiplied force.
Ideal gas: model where molecular volume and intermolecular interactions are neglected. Adequate for low-pressure gases far from liquefaction.
Density $ρ$ : mass per unit volume. Determines buoyancy, thrust and a thousand fluid behaviours.
Buoyancy force: does not depend on depth (in an incompressible, constant-density fluid), but on immersed volume and fluid density.

How to use it in the classroom

Opening: Pascal's Law tab. Show the two hydraulic cylinders with different sections. Apply a small force $F_{1}$ to the small piston: a much larger force is read on the large piston. Have students compute the ratio $A_{2} / A_{1}$ mentally and verify it matches $F_{2} / F_{1}$ . Guiding question: "Where does the extra energy come from? Is this perpetual motion?" (the answer is in the displacement: the small piston moves a lot, the large one little, the product $F \cdot s$ is equal).

Development: Boyle-Mariotte tab. Show the movable piston above a gas volume with particles in motion. Move the piston down (compression): the volume decreases, particles "bounce" against the walls more frequently, pressure rises. The $P$ - $V$ graph traces a hyperbola. Have students numerically verify that $P_{1} V_{1} = P_{2} V_{2}$ at two different points of the curve.

Deeper exploration: Archimedes tab. Select a material (e.g. wood in water): the object floats. Switch to iron in water: it sinks. Try aluminium in mercury: it floats. Highlight that the phenomenon depends only on the density ratio, not on shape, mass or size. Show the arrows $F_{A}$ and $P$ balancing (floating) or not.

Closing: real-world links. Final question: "A steel ship weighs hundreds of tons but floats. How is it possible, given that steel is much denser than water?" The answer leads naturally to the average density of the body, including the internal air volume.

Real-world examples

Hydraulic car brakes. The pedal drives a small piston that creates high pressure in the circuit; large pistons in the calipers exert large forces on the brake discs, pure Pascal's Law.
Hydraulic lifts and presses. Mechanical workshops, construction, heavy industry. A small manual or electric pump lifts tons.
Compressed gas cylinders. A "200 bar" cylinder actually contains a mass of gas equal to 200 times what would fit in the same volume at atmospheric pressure, direct application of Boyle.
Submarines and bathyscaphes. They vary their average density by filling or emptying water tanks to dive or surface.
Hot-air balloons. They float in air because the internal density (hot air, helium, hydrogen) is lower than that of the surrounding air.

Classroom discussion questions

In a hydraulic lift the small piston has 10 times less section than the large one. If I apply 50 N to the small one, what does the large one lift? And by how much does the small one descend to lift the large one by 1 cm?
I compress a gas to half its volume at constant temperature. What happens to the pressure? And if the volume drops to a tenth?
An object floats in fresh water. Moved to seawater (denser), does it float more or less? Why?
A gold block and an aluminium block have the same mass. Which one feels greater buoyancy when both are fully immersed in water? Why?
A cargo ship: how does the crew know if it has loaded too much? (hint: lines drawn on the side).

Related modules

Forces & Vectors: weight, buoyancy and pressure forces are vector quantities. Floating is a case of static equilibrium between two opposite vectors.