Physics notes / Fluids & Waves

Fluids & Waves · 100% marks notes

Fluids & Waves full marks = state assumptions and boundaries in ≤15s, then run Bernoulli/continuity, modulus, or standing-wave/Doppler templates inside 60–90 seconds.

Goal Bank ~10–14 /100 Phys (~full Fluids+Waves share)Weight ~12% of Physics tags (fluids ~2% + waves/SHM-sound ~10%, 2020–2026)Speed 60–90sShare 10.2%Trend Rising

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This portion is fluid statics/dynamics, elasticity/surface tension, and waves & sound. 100% means you stamp ideal/incompressible for Bernoulli, drop vs soap for ΔP, and nodes/antinodes before harmonics — never invent a formula mid-stem. Fluids & Waves reward assumption honesty and boundary clarity.

Topic-wise on this page

Fluids & Waves · chapter details

Same content as each topic page — brief, paper asks, methods, formulas, and families — so you can study here first, then open a full topic only when you need samples or trends.

01 · High-yield numericals once pressure, buoyancy and Bernoulli stay in separate lanes

Fluid Mechanics

Full topic page →

PromiseClassify hydrostatic vs flow in 15 seconds; one free-body or streamline equation plus a consistency check finishes most stems under 90 seconds.

Advanced mixes Pascal/manometer arithmetic with buoyancy paradoxes and Bernoulli applications. Never paste Bernoulli onto a viscous Stokes problem, and never treat gauge and absolute pressure as interchangeable. Fluid template: statics or flow? If flow—ideal or viscous? Ideal ⇒ continuity + Bernoulli on one streamline with assumptions; viscous ⇒ force balance, not Bernoulli.

01. Hydrostatics is force balance on a fluid element or immersed body: ΔP=ρgh along a vertical, and Pascal transmits pressure change undiminished in a confined incompressible liquid.

02. Buoyancy equals the weight of displaced fluid (Archimedes). Apparent weight and floating equilibrium are static free-body statements, not flow formulas.

03. For steady ideal flow along a streamline, continuity (A₁v₁=A₂v₂ for incompressible) and Bernoulli (P+½ρv²+ρgh=const) are energy/mass bookkeeping—not universal laws for every pipe.

How Advanced asks this

Manometer / hydraulic-press / force on a gate with Pascal and ρgh
Floating or fully submerged body: apparent weight, fraction immersed, two-liquid layers
Venturi, Torricelli, siphon or tank-hole speed from Bernoulli + continuity
Horizontal/vertical U-tube oscillation or pressure difference from height of liquid columns
Sphere falling in viscous liquid: terminal velocity, viscosity from timing, Stokes vs buoyancy
Multi-correct statements mixing streamline assumptions with viscosity or turbulence cues

Traps that leak marks

Using Bernoulli on a Stokes/viscous terminal-velocity stem
Confusing gauge and absolute pressure in manometers
Taking buoyancy with body density instead of fluid density
Applying A₁v₁=A₂v₂ to compressible or unsteady jets without care
Forgetting buoyancy when writing terminal-velocity force balance
Bernoulli on a Stokes/viscous terminal-velocity stem
Buoyancy with body density instead of fluid density
Confusing gauge and absolute pressure in manometers

Must cover (this topic)

Bernoulli applications
Stokes + terminal velocity

60–90s process

0–15s — Decide: hydrostatic pressure/Pascal, buoyancy free-body, ideal streamline flow, or Stokes/terminal.
15–35s — Draw free surface, datum, streamlines or FBD; write ΔP=ρgh, F_B=ρVg, continuity, Bernoulli, or ΣF=0 for terminal.
35–75s — Solve with consistent gauge/absolute choice and one density per fluid layer; cancel ρg when only height ratios matter.
75–90s — Check direction of pressure change with depth, floating fraction ≤1, and that Bernoulli was not used on a Stokes stem.

Methods that change the score

Hydrostatics FBDs and Bernoulli-with-stated-assumptions cover the high-yield fluid asks.

Method 01 · Column-by-column pressure walk

Manometer puzzles become arithmetic instead of memorised shapes.

Do this: Start from a known free surface or shared level and add/subtract ρgh for each vertical segment.

Method 02 · Displaced-volume free body

Floating and apparent-weight stems are pure statics once V_displaced is identified.

Do this: Write mg, F_B and any tensions/normal forces; never invent a Bernoulli term on a static body.

Method 03 · Streamline energy ledger

Venturi and Torricelli reduce to one algebraic difference of P, v and h.

Do this: Pick two points on one streamline; write Bernoulli once; insert continuity if areas change.

Method 04 · Terminal force triangle

Separates density difference from viscosity cleanly and blocks Poiseuille misuse.

Do this: At v_t set weight = buoyancy + Stokes drag (directions clear for rising bubble vs falling sphere).

Quick formulas

Hydrostatic gradient: ΔP=ρgh (vertical); P=P₀+ρgh below free surface — Incompressible liquid columns and manometers
Pascal: ΔP transmitted undiminished; F₁/A₁=F₂/A₂ (ideal hydraulic press) — Confined liquid force multiplication
Archimedes: F_B=ρ_fluid V_displaced g; W_app=W−F_B — Immersed or floating bodies
Floating: mg=ρ_fluid V_immersed g ⇒ V_imm/V=ρ_body/ρ_fluid — Equilibrium float in one fluid
Continuity: A₁v₁=A₂v₂ (incompressible, steady) — Same volume flux through sections
Bernoulli: P+½ρv²+ρgh=const along streamline (ideal, steady, incompressible) — Venturi, Torricelli, siphon
Torricelli: v=√(2gh) from free surface to hole (ideal) — Tank orifice with negligible surface speed
Stokes / terminal: F_d=6πηrv; at terminal: 6πηrv_t=(ρ−σ)V g — Small sphere in viscous fluid, Re≪1

Common question families

Hydrostatics and buoyancy

Looks like: Pressure depth, Pascal, floating/submerged

Fast model: P = P₀+ρgh; buoyant force = weight of displaced fluid.

Bernoulli + continuity

Looks like: Venturi, tank efflux, or streamline speed/pressure

Fast model: A₁v₁=A₂v₂ and P+ρgh+½ρv² constant along streamline.

Viscosity and terminal velocity

Looks like: Stokes drag, sphere in fluid

Fast model: At terminal speed, weight = buoyancy + drag.

Surface tension and capillarity

Looks like: Drop/bubble excess pressure, capillary rise

Fast model: ΔP = 2σ/r (drop) or 4σ/r (soap bubble); h=2σcosθ/(ρgr).

Sample preview · Hydrostatics and buoyancy

A wooden cube of relative density 0.8 floats in water. Find the fraction submerged.

Answer: 0.8

Floating balance equates weights of body and displaced liquid.

Properties of Matter

Full topic page →

PromiseTag the stem as surface/capillary, viscous drag, or Hookean elasticity in 15 seconds; one energy or force relation plus a unit check closes it.

This topic is three toolkits sharing one name. Advanced distractors swap excess pressure formulas between drops and bubbles, or confuse Young’s, shear and bulk moduli. Keep surface energy, viscous force and stress–strain in separate mental drawers. Matter template: elasticity vs surface vs viscosity → pick the defining equation → count surfaces / strain type → energy factor ½ when storing elastic energy.

01. Surface tension γ is force per unit length and also surface energy per unit area. Excess pressure inside a curved interface is set by geometry: 2γ/R for a liquid drop, 4γ/R for a soap bubble (two surfaces).

02. Angle of contact and capillary rise h=2γcosθ/(ρgr) encode wetting; meniscus shape follows whether the liquid wets the wall.

03. Viscosity η defines shear stress τ=η(dv/dx). Stokes drag on a sphere is in syllabus; laminar pipe Poiseuille flow is excluded—do not invent ΔP=8ηLQ/(πr⁴) on exam day.

How Advanced asks this

Work to blow a bubble / increase drop surface; excess pressure comparisons
Capillary rise/fall, angle of contact, two capillaries or liquid between plates
Viscous force, velocity gradient, Stokes terminal (linked to fluids) without Poiseuille
Wire stretch, shear of a block, bulk compression; Y, η, B and Poisson links
Statement sets on Hooke’s law limits, elastic energy, or which modulus applies
Integer/numerical on force along a film of length L or energy of n bubbles coalescing

Traps that leak marks

Using 2γ/R for a soap bubble
Forgetting cosθ or sign in capillary rise
Inserting Poiseuille for any viscosity stem
Using Young’s modulus for pure shear or bulk compression
Omitting ½ in elastic energy
Taking film length as diameter instead of total length in contact
Using 2γ/R for a soap bubble
Omitting ½ in elastic energy
Forcing Poiseuille onto any viscosity mention

Must cover (this topic)

Surface tension / capillary
Young / rigidity / bulk modulus ideas

60–90s process

0–15s — Classify: surface/excess pressure/capillary, viscosity/Stokes, or elastic modulus (Y / shear / bulk).
15–35s — Write γ definitions, ΔP=2γ/R or 4γ/R, h=2γcosθ/(ρgr), τ=η dv/dx, or stress=modulus×strain.
35–75s — Count surfaces carefully; use consistent SI for γ, η and moduli; integrate force×distance only for work against tension.
75–90s — Check drop vs bubble factor, cosθ sign for rise/fall, and that the correct modulus matches the deformation type.

Methods that change the score

Stress–strain definitions and constraint expansions unify elasticity numericals.

Method 01 · Surface-count stamp

Kills the classic 2γ/R vs 4γ/R trap instantly.

Do this: Before ΔP or work, write one surface (drop) or two (soap film/bubble).

Method 02 · Capillary force–weight balance

Derives h without memorising sign errors; θ>90° gives depression.

Do this: Vertical component 2πr γ cosθ = weight of raised column ρ(πr²h)g.

Method 03 · Deformation-type → modulus

Prevents plugging shear data into Young’s formula.

Do this: Match change in length → Y, angle/shape → rigidity, volume → B.

Method 04 · Energy-density route

Matches Hookean linearity and catches factor-½ omissions.

Do this: For stretch work, prefer U=½×stress×strain×volume over inventing average forces.

Quick formulas

Surface tension: γ=F/L; also γ=surface energy/area — Force on film or energy to create surface
Excess pressure: ΔP=2γ/R (drop/air bubble in liquid); ΔP=4γ/R (soap bubble) — Curved interface pressure jump
Capillary rise: h=2γcosθ/(ρ g r) — Narrow tube; θ=angle of contact
Viscous shear: τ=η (dv/dx); F=ηA(dv/dx) — Laminar layers; definition of η
Stokes drag: F=6πηrv — Small sphere, low Re (no Poiseuille)
Young’s modulus: Y=(F/A)/(ΔL/L) — Longitudinal stretch/compression
Rigidity / bulk: η=(F/A)/θ; B=−ΔP/(ΔV/V) — Shear angle θ; volume compression
Elastic energy: U=½ F ΔL=½ (stress)(strain)(volume) — Hookean stored energy

Common question families

Young’s modulus and stress–strain

Looks like: Wire elongation, Y from load–extension

Fast model: Y = (F/A)/(ΔL/L); energy = ½×stress×strain×volume.

Bulk and shear modulus

Looks like: Volume compression or angle of shear

Fast model: B = −ΔP/(ΔV/V); η = shear stress / shear strain.

Elastic potential energy

Looks like: Work to stretch a wire or spring-like rod

Fast model: U = ½ × stress × strain × volume = ½ F ΔL.

Composite / thermal–elastic couple

Looks like: Bimetallic or rod with mismatch under load/temperature

Fast model: Mechanical ΔL plus thermal ΔL must match constraints.

Sample preview · Young’s modulus and stress–strain

A wire of length 1 m and area 1 mm² stretches 1 mm under 100 N. Find Y.

Answer: 10¹¹ Pa

Convert area to m² before dividing.

Waves & Sound

Full topic page →

PromiseName wave type and boundary conditions first; then superposition, harmonic formula or Doppler shift finishes the stem without formula thrash.

Advanced loves stationary waves on strings and in air columns, then dresses them with beats or Doppler. Progressive-wave kinematics and intensity are the setup language—boundary conditions are where marks are won or lost. Wave template: progressive vs standing → boundaries → v from medium → f,λ from v=fλ → Doppler/beats only after geometry is drawn.

01. A plane progressive wave carries phase at speed v=fλ; transverse vs longitudinal is about particle motion relative to propagation, not about whether sound can exist in solids.

02. Superposition is linear addition of displacements. Beats: |f₁−f₂| intensity maxima per second when two close frequencies meet.

03. Stationary waves form from opposite travelling waves: nodes fixed, antinodes max amplitude. String harmonics need both ends’ boundary conditions; organ pipes need open/closed end rules.

How Advanced asks this

Equation of progressive wave: amplitude, λ, f, v, phase difference
String with both ends fixed / free–fixed variants: harmonic spectrum
Closed/open pipe: end correction, resonance with tuning fork, temperature shift
Beats between two strings/forks; load or tension change to match frequency
Doppler: source/observer/reflector moving; apparent frequency and beat with echo
Intensity, loudness, or phase/path difference interference (when linked to sound)

Traps that leak marks

Even harmonics in an ideal closed pipe
Swapping source and observer Doppler factors
Using light-Doppler formula for sound
Forgetting end correction when the stem gives it
Confusing progressive phase speed with particle speed ωA
Taking beat frequency as (f₁+f₂)/2
Even harmonics in an ideal closed pipe
Swapping source and observer Doppler factors
Using light-Doppler for sound

Must cover (this topic)

Standing waves on strings/air columns
Beats; Doppler in sound
Superposition of plane waves

60–90s process

0–15s — Classify: progressive parameters, stationary string/pipe, beats, sound-speed, or Doppler geometry.
15–35s — Sketch nodes/antinodes or source–observer–wind arrows; write v=fλ, harmonic f_n, v_sound, or Doppler formula with signed speeds.
35–75s — Apply boundary conditions before picking n; keep beat frequency as |Δf|; convert temperature/medium changes through √T.
75–90s — Check which end is displacement node, Doppler sign (approach raises f), and that string μ and T match v=√(T/μ).

Methods that change the score

Boundary sketches and a Doppler sign convention prevent harmonic and frequency mistakes.

Method 01 · Boundary sketch first

Stops even-harmonic answers on closed pipes and wrong λ on strings.

Do this: Mark fixed→node, free/open→antinode before writing f_n.

Method 02 · Harmonic ladder

Tuning-fork + variable-length column stems become counting, not guessing.

Do this: Write allowed f as multiples of fundamental; match given resonance to the correct rung.

Method 03 · Beat as frequency dial

Resolves ambiguous ‘beat decreases when tension increases’ statements.

Do this: Treat f_beat as |f−f₀|; ask whether tension/length change raised or lowered f.

Method 04 · Doppler arrow audit

Sign errors vanish when geometry is drawn once.

Do this: Draw v arrows for source, observer, wind; apply approach ⇒ numerator up / denominator down.

Quick formulas

Wave basics: v=fλ; ω=2πf; k=2π/λ; y=A sin(ωt−kx+φ) — Progressive plane wave parameters
String speed: v=√(T/μ); f_n=n v/(2L) (both ends fixed) — Stretched string harmonics
Open pipe: f_n=n v/(2L), n=1,2,3… (ideal) — Both ends displacement antinodes
Closed pipe: f_n=(2n−1)v/(4L), n=1,2,3… — One end closed: odd harmonics only
Beats: f_beat=|f₁−f₂| — Two close frequencies heard together
Sound in gas: v=√(γP/ρ)=√(γRT/M) — Ideal gas; γ=C_P/C_V
Doppler (sound): f'=f (v±v_o)/(v±v_s) (signs: toward each other → higher f) — Source/observer moving through still air
Phase / path: Δφ=(2π/λ)Δx — Interference path difference

Common question families

Progressive wave parameters

Looks like: Read ω, k, v, phase from y(x,t)

Fast model: Match to A sin(ωt∓kx+φ); v=ω/k.

Standing waves on string / pipe

Looks like: Harmonics, nodes, end conditions

Fast model: Sketch boundary nodes/antinodes; f_n = nv/(2L) or odd harmonics.

Beats and superposition

Looks like: Two close frequencies or intensity interference (sound)

Fast model: f_beat = |f₁−f₂|; resultant amplitude depends on phase.

Doppler effect (sound)

Looks like: Moving source/observer, wind

Fast model: Use f′ = f (v±v_o)/(v±v_s) with sign convention stated once.

Sample preview · Standing waves on string / pipe

A string of length 1 m fixed at both ends vibrates in its first overtone with wave speed 40 m/s. Find the frequency.

Answer: 40 Hz

Count harmonics from boundary conditions.

How to use these notes for 100%

Point 01

Depth upgrade: Bernoulli needs assumptions (steady, incompressible, along streamline). Name them before equating P+ρgh+½ρv².

Point 02

Waves/SHM: separate oscillator energy language from standing-wave boundary conditions — do not teach both as one blur.

Speed system

60–90 second answer protocol

Use this on every stem — SC, MC, NUM, MATCH.

0
0–15s — Classify: hydrostatic/buoyancy, Bernoulli/continuity, viscosity/terminal, elasticity/surface, progressive/standing wave, beats, Doppler. Mark densities, open/closed ends, and what is asked.
15
15–35s — Write assumptions (steady, inviscid, incompressible) or boundary sketch (nodes/antinodes). Pick the governing equation set only.
35
35–75s — Execute template: P=ρgh + Archimedes, Bernoulli+continuity, Y/B/η definitions, f_n from boundaries, or Doppler with a fixed sign convention.
75
75–90s — Check gauge vs absolute pressure, drop vs soap surfaces, open vs closed harmonic set, Doppler signs. Bubble.

Prove each statement from the stamped assumptions or boundary conditions — never from another option.

Why it saves time

Protects clusters of marks on wave MC.

Instant recall

Mental bank

Keep these cold — no derivation during the 60–90s window.

Hydrostatic: P=P₀+ρgh; buoyant force = ρ_fluid V_imm g
Bernoulli: P+ρgh+½ρv² = const (ideal streamline)
Continuity: A₁v₁=A₂v₂ (incompressible)
Stokes terminal: 6πηrv = (ρ−σ)V g
ΔP drop = 2σ/r; soap bubble = 4σ/r; capillary h=2σ cosθ/(ρgr)
Y=(F/A)/(ΔL/L); U=½ F ΔL for gradual load
String fixed ends: f_n = n v/(2L); closed pipe: odd harmonics
Doppler: f′=f(v±v_o)/(v±v_s); beats |f₁−f₂|
Bernoulli assumptions spoken first
Open/closed pipe harmonics from boundary
Doppler: which source/observer moves

Inside Fluids · Waves

Sub-portion playbooks

Each family has a trigger, template, and drill.

01 · Lower frequency — high conversion if templated

Fluid Mechanics

45s triggerSee pressure/float/Venturi/viscosity → hydrostatic FBD or Bernoulli assumptions.

Template (write on rough)

Statics: pressure map + Archimedes FBD.
Dynamics: stamp assumptions; continuity; Bernoulli.
Viscosity: weight = buoyancy + Stokes drag at terminal speed.
MC: check gauge/absolute and ideal-flow claims separately.

Must automate

Fraction submerged = ρ_body/ρ_liquid (floating)
Torricelli v=√(2gh) with surface-speed caveat
Terminal velocity force balance

Traps

Using Bernoulli through a viscous valve without stating loss
Wrong displaced volume for partially immersed bodies

Drill

3×/week: 1 buoyancy + 1 Bernoulli NUM

02 · Medium — elasticity + capillarity

Properties of Matter · Surface Tension

45s triggerSee wire stretch, ΔV, shear, drop/bubble, capillary → modulus or surface count.

Template (write on rough)

Identify Y vs B vs η.
Write defining ratio; solve ΔL, ΔV, or θ.
Surface: drop (2σ/r) vs soap (4σ/r); capillary with cosθ.
Energy: include ½ for elastic; γΔA for surface.

Must automate

Y numerical with area in m²
Bulk modulus from % volume change
Drop vs soap excess pressure

Traps

Soap counted as one surface
Thermal–elastic mismatch solved without length constraint

Drill

Weekly: 2 modulus + 1 capillarity/surface stem

03 · High inside this portion

Waves & Sound

45s triggerSee y(x,t), harmonics, beats, Doppler → match form or boundary sketch.

Template (write on rough)

Progressive: read A, ω, k, direction; v=ω/k.
Standing: boundaries → λ_n → f_n.
Beats: |f₁−f₂|; amplitude bounds if asked.
Doppler: fix signs; double application for reflection.

Must automate

Open vs closed pipe spectra
End correction elimination via successive resonances
Doppler echo / moving-wall pattern

Traps

Even harmonics on a closed pipe
Inconsistent Doppler signs between echo legs

Drill

Daily: 1 standing-wave + 1 Doppler/beats stem

Habit

Common practice system

Same system every week — builds transferable speed.

Daily

20 min: 3 Fluids/Waves stems — write assumptions or boundary sketch by 15s.
Force independent T/F on every multi-correct.
Log any stem where Doppler signs were flipped mid-solution.

Weekly

1 organ-pipe + end-correction timed set.
1 Bernoulli/buoyancy mixed pair.
Rebuild drop/soap/capillary sheet from memory.

Before every mock

Flash: Bernoulli assumptions, harmonic formulas, Doppler signs — 5 minutes.
In paper: bank direct wave-parameter and buoyancy asks early.

Full marks gate

100% checklist for Fluids & Waves

Tick these only when true on timed practice — not when “you know the chapter”.

Bernoulli stems always state ideal-flow assumptions.
Buoyancy uses displaced-fluid weight, not body volume blindly.
Modulus type identified before formula.
Drop vs soap surface count is explicit.
Standing-wave boundaries sketched before f_n.
One Doppler sign convention used throughout.
Every multi-correct statement gets an independent verdict.
I never apply Bernoulli across a pump or unsteady region blindly.
Standing-wave nodes/antinodes match the boundary I named.

Paper coverage for Fluids · Waves Syllabus map Subject analysis Physics · 300 plan

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