5.4 Fluids, Pressure, Flow & Basic Electrical Circuits

Fluids, pressure, flow and basic electrical circuits

Pressure and force

Pressure is force distributed over area:

P = F/A.

The pascal is one newton per square metre. The same force on a smaller area produces greater pressure. The same pressure over a larger area produces greater total force.

In an ideal enclosed hydraulic system, Pascal’s principle gives equal transmitted pressure:

F1/A1 = F2/A2.

If an input piston has area 5 cm² and an output piston has area 100 cm², the area ratio is 20. An input of 150 N ideally produces 3,000 N at the output. The output piston moves only one twentieth as far as the input piston for equal displaced volume, preserving ideal work.

Real systems lose energy through friction, leakage, fluid heating, and deformation. Aptitude diagrams often state that these are ignored.

Hydrostatic pressure

In a stationary fluid of density ρ, pressure change with vertical depth h is:

ΔP = ρgh.

Pressure depends on vertical depth, density, and gravity—not the container’s shape. At the same depth in the same connected fluid, pressure is the same under ideal static conditions.

A point deeper below a free surface has greater gauge pressure. If two tanks contain different liquids to the same depth, the denser liquid creates greater pressure at the bottom.

Flow rate and continuity

Volume flow rate is:

Q = volume/time = Av,

where A is cross-sectional area and v is average flow speed. For steady incompressible flow through one path:

A1v1 = A2v2.

If area halves while flow rate remains constant, speed doubles. Diameter requires care because circular area is proportional to diameter squared. Halving diameter makes area one quarter, so speed must become four times as large for the same flow rate.

This relationship alone does not say pressure always falls by a particular amount; complete pressure predictions may require height, energy losses, pumps, and boundary conditions. Use only the assumptions supplied.

Buoyancy and density

An immersed object experiences an upward buoyant force equal to the weight of displaced fluid. It floats in equilibrium when buoyant force equals its weight. An object less dense than the fluid can float with part above the surface; a denser unsupported object sinks.

Density is mass per volume. A 2 kg object with volume 0.001 m³ has density 2,000 kg/m³. Compare with the stated fluid density rather than relying on material appearance.

Circuit foundations

Voltage is electric potential difference, current is rate of charge flow, and resistance opposes current. Ohm’s law for an ohmic component is:

V = IR.

Rearrange to I = V/R or R = V/I. With 12 V across 6 Ω, current is 2 A.

Electrical power is:

P = VI.

Combining with Ohm’s law gives P = I²R and P = V²/R when the assumptions fit.

Series circuits

Components in one path are in series.

• Current is the same through every component.
• Voltage drops add to the supply voltage.
• Resistances add: R total = R1 + R2 + ...
• Opening the single path stops current everywhere.

For 4 Ω and 8 Ω in series across 24 V, total resistance is 12 Ω and current is 2 A. Voltage drops are 8 V and 16 V, which total 24 V.

Parallel circuits

Parallel branches connect across the same two nodes.

• Voltage is the same across each branch.
• Branch currents add to total current.
• Reciprocal resistances add: 1/R total = 1/R1 + 1/R2 + ...
• Equivalent resistance is less than the smallest branch resistance.

For 6 Ω and 3 Ω in parallel:

1/R = 1/6 + 1/3 = 1/2, so R total = 2 Ω.

Across 12 V, branch currents are 2 A and 4 A, totalling 6 A. The total also equals 12 V ÷ 2 Ω.

Switches and short circuits

An open switch breaks its path. In a series circuit it stops all current; in one branch of a parallel circuit it stops only that branch while other complete branches can remain active.

An ideal short circuit has extremely low resistance and can draw very large current. Never infer that adding a parallel branch increases total resistance; it creates another path and lowers equivalent resistance.

Diagram method and checks

For fluids, label area, speed, height, density, and flow direction. For circuits, mark nodes: components connected to the same two nodes are parallel even if the drawing is bent. Predict first: a larger hydraulic output area multiplies force; a narrower constant-flow pipe increases speed; adding series resistance reduces current for fixed voltage; adding a parallel branch increases total current for fixed voltage. Then calculate and check units.