Apex Class Logo
APEX CLASS ESTD 2018

IITJEE: Bernoulli’s Theorem, Torricelli’s Efflux & Surface Tension Visualizer

By

Ashish Kumar Bahekar

Published on

in

Apex Class • IITJEE Fluid Mechanics & Surface Tension Lab

Bernoulli’s Theorem, Torricelli’s Efflux & Surface Tension Visualizer

Master Venturi Pressure Drops, Torricelli Water-Jet Range, Stokes’ Terminal Speed, and Soap Bubble Excess Pressure!

🔬 Select a Fluid Dynamics or Viscosity Setup:
↕ Δh (P₁ – P₂ = ρgΔh) Area A₁, Slow v₁ High Pressure P₁ Narrow A₂, Fast v₂! Low Pressure P₂ Continuity & Bernoulli Equations 1. Continuity: A₁ v₁ = A₂ v₂ (v ∝ 1/r²!) 2. Bernoulli: P + ½ ρ v² + ρ g h = Const 3. Horizontal Pipe: P₁ – P₂ = ½ ρ (v₂² – v₁²) 4. Venturi Flow Speed: v₁ = A₂ √[ 2gΔh / (A₁² – A₂²) ] ✈️ Golden Bernoulli Rule: Where streamline velocity is HIGH, static pressure is LOW (Aeroplane Wing Lift & Magnus Effect)! Water Level (H) ↕ H Same R for h & (H-h)! Midpoint (h = H/2) 🏆 R_max = H! Torricelli’s Efflux Master Formulas 1. Efflux Speed: v = √(2 g h) 2. Time to Hit Ground: t = √[2(H – h)/g] 3. Horizontal Range: R = 2 √[ h(H – h) ] 4. Max Range at h = H/2 ⇒ R_max = H! • Reaction Thrust on Tank: F = ρ a v² = 2ρgah 🚰 Complementary Depth Trick: Holes at depth ‘h’ from top and ‘h’ from bottom ALWAYS strike the ground at the exact same point! Sphere (ρ) Weight W = (4/3)πr³ρg ↓ ↑ Buoyancy F_B ↑ Drag F_v = 6πηrv_T Stokes’ Law & Terminal Velocity (F_net = 0) v_T = [ 2 r² (ρ – σ) g ] / (9 η) • Key Scaling: v_T ∝ r² (If radius doubles → v_T is 4×!) • N Drops Coalesce Trick (R = N¹/³ r): v_big = N²/³ × v_small (For N = 8 drops → v_big = 4 v_small!) • Rate of Heat Produced at v_T: dH/dt = F_v · v_T ∝ r⁵! 🌧️ Air Bubble in Water Trap: If sphere density ρ < liquid density σ, (ρ – σ) is negative → Bubble rises UPWARD at v_T! 1. Archimedes’ Principle & Floating Ice • Fraction Submerged: V_in / V_total = ρ_body / ρ_liquid • Apparent Weight: W_app = W₀ (1 – ρ_liquid / ρ_body) • Ice Melting in Pure Water: Level stays UNCHANGED! • Ice with Lead Ball Inside Melts: Water level DROPS! 2. Hydrostatic Pressure & U-Tube Laws • Gauge Pressure at Depth h: P = P₀ + ρ g h • Immiscible U-Tube: ρ₁ h₁ = ρ₂ h₂ (h ∝ 1/ρ) • Accelerating Container (a_x horiz, a_y up): Free Surface Tilt Angle: tanθ = a_x / (g + a_y)! ⚖️ Pascal’s Hydraulic Lift Law: F₁ / A₁ = F₂ / A₂ (Multiplies Force by A₂/A₁, while Work Done W₁ = W₂ stays conserved)!
Volume Flow Rate (Q)
Q = A · v = Constant
SI Unit: m³ / s (Incompressible)
Torricelli Range Formula
R = 2 √[ h(H – h) ]
Max Range R_max = H at h = H/2
Terminal Velocity (v_T)
2r²(ρ – σ)g / (9η)
v_T ∝ r² | v_big = N²/³ v_small
Poiseuille’s Capillary Flow
Q = π ΔP r⁴ / (8 η L)
Fluid Resistance R_f = 8ηL/(πr⁴)
🫧 Surface Tension T = F/L = ΔU/ΔA. Select a High-Yield JEE Problem Type:
1. Water Drop / Air Bubble in Water (1 Surface) P_in > P₀ ΔP = P_in – P₀ = 2T / R • Surface Area: A = 4π R² • Surface Energy: U = 4π R² T (Only 1 liquid-air interface!) 2. Soap Bubble in Air (2 Free Surfaces!) Air Inside ΔP = P_in – P₀ = 4T / R! • Effective Area: A_eff = 2 × (4π R²) = 8π R² • Surface Energy: U = 8π R² T (Double!) (Air touches both inside & outside of soap film!) 🎯 JEE Trap: An air bubble inside water has ONLY ONE surface (ΔP = 2T/R), whereas a soap bubble in air has TWO (ΔP = 4T/R)! Small r₁ (High P₁!) Large r₂ (Low P₂) Interface Bulges into Larger Bubble (r₂)! 🏆 3 Classic Two-Bubble JEE Problems 1. Common Interface Radius (r₁ < r₂): 4T/R = 4T/r₁ – 4T/r₂ ⇒ R = (r₁ r₂) / (r₂ – r₁) 2. Isothermal Coalescing in Vacuum (PV = Const): R_new = √(r₁² + r₂²) 3. Connected by Tube: Small bubble SHRINKS, Big GROWS! ⚠️ Connected Bubbles Trap: Since ΔP = 4T/r, the SMALLER bubble has HIGHER internal pressure and blows air into the larger bubble! 1 Big Drop Radius R Initial U_i = 4π R² T Splits into N Final U_f = N¹/³ (4π R² T) Droplet Splitting & Coalescing Formulas 1. Volume Conservation: R = N¹/³ · r 2. Work Done in Splitting (Area Increases): W = ΔU = 4π R² T (N¹/³ – 1) = 4π r² T (N – N²/³) 3. Temperature Change when N Drops Coalesce: Δθ = [ 3 T / (ρ s) ] (1/r – 1/R) (Heats Up!) 💥 Quick Check: Splitting 1 drop into N droplets ABSORBS energy (Cools down); Coalescing N drops RELEASES energy (Heats up)! ↕ Rise h Concave Meniscus (θ < 90°) Jurin’s Law & Capillary Rise Shortcuts h = (2 T cosθ) / (r ρ g) = (2 T) / (R_meniscus ρ g) • Pure Water on Clean Glass: θ = 0° ⇒ cos0° = 1 • Mercury on Glass: θ ≈ 135° (>90°) ⇒ Capillary Fall! • Tube of Insufficient Length (L < h): Water NEVER overflows! Meniscus flattens: h · R = L · R’! 🧪 Satellite / Free-Fall Trap: In a satellite (g_eff = 0), water rises to the FULL length of the tube and flattens out without overflowing!
Excess Pressure (ΔP)
Drop: 2T/R | Bubble: 4T/R
Concave side always has higher P!
Double Bubble Radius
R = (r₁ r₂) / (r₂ – r₁)
Vacuum Coalesce: R = √(r₁² + r₂²)
Capillary Rise (h)
h = 2T cosθ / (r ρ g)
Jurin’s Law: h · r = Constant!
Heat Lost in Capillary Rise
ΔH = ½ m g h
Work by Surface Tension = 2 × ΔU_grav!
🔥 2 Bonus Elasticity & Surface Tension Shortcuts for JEE Main:
1. Elastic Potential Energy Density in a Stretched Wire: u = U / Volume = ½ × Stress × Strain = ½ Y (ΔL / L)² (Total Energy U = ½ F · ΔL).
2. Effect of Temperature & Detergent on Surface Tension: Raising temperature or adding soap/detergent DECREASES Surface Tension T (At Critical Temperature T_c, Surface Tension becomes ZERO)!
, , ,

Read Next:

Leave a Reply

Apex Class App

Play revision games, read notes, and test your skills anywhere!

🚀

Fresh Updates

Latest notes, lectures & announcements!

View All Posts ➔
⚡

Interactive Zones

🎮
Study Games Arena Play & Revise Concepts
→ 🎯
Study Quiz Arena Test Your Knowledge
→ 🏛️
UPSC Quiz Arena Master Civil Services
→

Discover more from Apex Class

Subscribe now to keep reading and get access to the full archive.

Continue reading

Enable Notifications OK No thanks