“Bubble has two skins”
Why the bubble formula has a 4
A soap bubble has an inner and an outer surface, so its excess pressure 4T/r is twice a liquid drop's 2T/r.
Class 11 Physics · Gujarat Board, CBSE, GUJCET and JEE
Mechanical Properties of Fluids formulas
Fluids is three separate chapters joined together: pressure, flow, and surface tension with viscosity. Each part has its own small set of formulas, and questions almost never mix them, so it is easier than its length suggests.
20 formulas, 4 memory hooks, and the 5 mistakes that cost the most marks.
Pressure and floating
| What | Formula | Symbols and notes |
|---|---|---|
| Pressure in a liquid | p = ρgh | depends on depth, never on the container's shape |
| Absolute pressure | p = p₀ + ρgh | p₀ is atmospheric pressure |
| Pascal's law | pressure applied is transmitted undiminished | the basis of the hydraulic lift |
| Hydraulic lift | F₂ = F₁(A₂/A₁) | a small force on a small piston lifts a large load |
| Buoyant force | FB = ρliquidVg | V is the volume submerged, not the whole body |
| Floating condition | ρbody/ρliquid = fraction submerged | ice floats with about 90% under water |
Flow
| What | Formula | Symbols and notes |
|---|---|---|
| Equation of continuity | A₁v₁ = A₂v₂ | a narrower pipe means faster flow |
| Bernoulli's theorem | p + ½ρv² + ρgh = constant | for steady, non-viscous, incompressible flow |
| Speed of efflux | v = √(2gh) | Torricelli's result, the same as a free fall through h |
| Venturi meter | v₁ = A₂√(2(p₁ − p₂)/ρ(A₁² − A₂²)) | used to measure flow speed |
Viscosity
| What | Formula | Symbols and notes |
|---|---|---|
| Newton's law of viscosity | F = −ηA(dv/dx) | η is the coefficient of viscosity, unit Pa s |
| Stokes' law | F = 6πηrv | drag on a small sphere moving slowly |
| Terminal velocity | vt = 2r²(ρ − σ)g/9η | ρ sphere density, σ fluid density |
| Reynolds number | Re = ρvD/η | below about 2000 the flow is streamlined, above 3000 turbulent |
| Poiseuille's equation | Q = πpr⁴/8ηL | the r to the fourth is why narrow arteries matter so much |
Surface tension
| What | Formula | Symbols and notes |
|---|---|---|
| Surface tension | T = F/L | force per unit length, unit N m−1 |
| Excess pressure in a drop | Δp = 2T/r | one surface |
| Excess pressure in a bubble | Δp = 4T/r | two surfaces, so twice the drop |
| Capillary rise | h = 2T cosθ/(rρg) | θ is the angle of contact; narrow tube, higher rise |
| Work to make a drop | W = T × ΔA | the increase in surface area times the surface tension |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
Know the formula but not when to use it?
Darshit Sir has taught this in Paldi since 2006, in the classroom and live online.
Memory hooks
How to remember this chapter.
“Narrow means fast means low pressure”
Continuity plus Bernoulli
A narrower pipe forces the liquid faster, and faster flow means lower pressure. That one chain explains aeroplane lift, the spin on a cricket ball and the atomiser.
“Submerged volume, not total”
Buoyancy
The upthrust uses the volume of liquid displaced. For a floating body that is only the part under the surface.
“Terminal when drag equals weight”
Terminal velocity
A falling drop speeds up until viscous drag plus upthrust balances weight. After that the speed is constant and the acceleration is zero.
Where marks go
The mistakes that cost the most.
- Using the whole volume in the buoyancy formula for a floating body. Only the submerged part counts.
- Thinking pressure depends on the shape or width of the vessel. It depends only on depth, density and g.
- Using 2T/r for a soap bubble. A bubble has two surfaces, so it is 4T/r.
- Applying Bernoulli's theorem to a viscous or turbulent flow. It assumes neither.
- Forgetting that terminal velocity is proportional to r², so a drop twice as wide falls four times as fast.
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