Vedatom Physics Mechanics / Fluid Mechanics
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Reading fluids right
Pressure depends only on depth. P = P₀ + ρgh — the shape of the vessel and the amount of liquid above you don't matter, only how far down you are.
Pascal transmits pressure. A pressure applied to an enclosed fluid shows up undiminished everywhere — the whole basis of the hydraulic lift.
Buoyancy = weight of fluid displaced. You float when your average density is below the fluid's, submerged just enough that the pushed-aside fluid weighs as much as you do.
Fast fluid, low pressure. Along a streamline P + ½ρv² + ρgh stays constant, so where the flow speeds up the pressure drops.
Accelerate the frame, tilt the surface. A fluid pushed sideways levels off perpendicular to g_eff = √(g²+a²); tan θ = a/g.
Efflux is free fall. A jet from depth h leaves at √(2gh) and lands farthest when the hole sits at mid-depth.
run the Venturi bench · §04
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§02

Pascal's principle & the hydraulic press

signature bench

Squeeze an enclosed fluid and the pressure it feels is the same everywhere. A small force on a small piston makes that pressure; a large piston turns the same pressure back into a large force. The catch is honest bookkeeping — the big piston barely moves, so the work you put in equals the work you get out.

{{ hy.label }}
F₁ = {{ hyF }} N small piston · A₁ load F₂ = {{ hy.F2 }} N large piston · A₂
input force F₁{{ hyF }} N
area ratio A₂A₁{{ hyA }} : 1
output force F₂ = F₁ · A₂A₁
{{ hy.F2 }} N
mechanical advantage
× {{ hyA }}
common pressure
{{ hy.P }}
Push the input down by {{ hyA }} cm and the load rises just 1 cm — the force is multiplied ×{{ hyA }}, the distance divided by {{ hyA }}. No free energy.
§01

Pressure grows with depth, not with volume

Every layer of fluid has to hold up all the fluid above it, so pressure climbs steadily as you descend: P = P₀ + ρgh. Only the depth h and the density ρ matter — a narrow straw and a vast lake reach the same pressure at the same depth. Change the liquid, lower the probe, and read the gauge.

Bench · a pressure probe in a tank
surface · P₀ = 1 atm h = {{ prDepth }} m probe · P acts all ways
depth below surface{{ prDepth }} m
gauge pressure · ρgh
{{ pr.gauge }} kPa
absolute pressure · P₀ + ρgh
{{ pr.abs }} kPa · {{ pr.atm }} atm
depth (m) P
{{ pr.note }}
§03

Archimedes — float, sink, or hang

A submerged body feels an upward push equal to the weight of the fluid it displaces: F_B = ρfluid Vsub g. Float when your average density sits below the fluid's, sinking only until the displaced fluid weighs as much as you. Dial the block's density and swap the liquid to see where it settles.

Bench · a 1 m³ block in a tank
waterline F_B mg
block density ρ_body{{ buDensity }} kg/m³
fraction submerged
{{ bu.subPct }}%
= ρbody ρfluid
buoyant force
{{ bu.FB }}
weight mg
{{ bu.W }}
{{ bu.verdict }} {{ bu.note }}