2.2.8
Fluid ka column ek pressure-measuring stick hai. Gravity fluid ko neeche kheenchti hai, isliye jitna gehre jaoge, utna zyada fluid ka weight tumhare upar hoga — aur wahi extra weight hi extra pressure hai. Manometer/barometer bas humein pressure ko force ki jagah height h padhne deta hai.
Intuition Height pressure kyun batati haiEk lamba liquid column imagine karo. Depth h badhna hi padega taaki upar ka column support ho sake. Jitna badhaav hota hai woh exactly fluid ka weight per unit area hota hai. Yahi saara raaz hai.
Definition Hydrostatic setupFluid static equilibrium mein hai (hil nahi raha). Density ρ g
Ek fluid cylinder lo : cross-sectional area A h 1 h 2 h 2 > h 1
Top par pressure neeche dhakelta hai: P 1 A
Bottom par pressure upar dhakelta hai: P 2 A
Weight neeche kheenchta hai: W = m g = ρ V g = ρ A ( h 2 − h 1 ) g
Yeh step kyun? Equilibrium mein net force zero hota hai, toh upar-wale forces = neeche-wale forces.
P 2 A = P 1 A + ρ A ( h 2 − h 1 ) g
A kyun? kyunki pressure, force per area hai, toh area kabhi relationship mein matter nahi karta):
P 2 = P 1 + ρ g ( h 2 − h 1 )
Mercury se bhari ek tube, dish mein ulti rakhi, upar vacuum banakar. Atmospheric pressure mercury ko upar dhakelta hai; column height h P a t m
Derivation. Do points lo jo same height par hain: point A tube ke andar dish surface par, point B open dish surface par.
B par (open surface): pressure = P a t m
A par (same level, tube ke andar): upar column ka pressure = + ρ g h = 0 + ρ g h
Equal kyun? Same fluid, same horizontal level ⇒ same pressure (warna fluid sideways flow karne lagta).
P a t m = ρ H g g h
Worked example Standard atmosphereρ H g = 13600 kg/m 3 g = 9.8 h = 0.76 m P a t m = 13600 × 9.8 × 0.76 ≈ 1.013 × 1 0 5 Pa Yeh step kyun? Measured column height ko ρ g h 760 mmHg = 1 atm hai. ✓
Intuition Mercury kyun, water kyun nahi?Water ko 1 atm balance karne ke liye ≈ 10.3 m h = P / ρ g ρ h 13.6 × 13.6 × 0.76 m
Definition Open-tube manometerEk U-tube jo partially liquid se bhari hai. Ek arm ek gas/vessel se connect hai jiska pressure P g a s height difference h
Derivation. Liquid ka sabse neecha common level lo — use reference line kaho. Is line par pressure dono arms mein same hai (connected, same fluid, same height).
Left arm (gas side), gas se line tak neeche jaate hue: P l e f t = P g a s + ρ g h g a s co l
Right arm (open side): P r i g h t = P a t m + ρ g h o p e n co l
P l e f t = P r i g h t h dono free surfaces ka difference ho:
P g a s = P a t m + ρ g h
jahan h positive hota hai agar open arm upar ho (gas apni side ko neeche dhakelta hai).
Worked example Worked manometerOpen-tube mercury manometer: open arm liquid h = 0.05 m upar hai gas arm se. P a t m = 1.01 × 1 0 5 P ab s = P a t m + ρ g h = 1.01 × 1 0 5 + 13600 ( 9.8 ) ( 0.05 ) = 1.01 × 1 0 5 + 6664 ≈ 1.077 × 1 0 5 Pa Yeh step kyun? Open arm upar ⇒ gas pressure atmospheric se zyada ⇒ ρ g h jodo . Gauge pressure = 6664
Worked example Gas atmospheric se neecheAb gas arm 0.05 upar hai (atmosphere gas side ko neeche dhakelta hai):
P ab s = P a t m − ρ g h = 1.01 × 1 0 5 − 6664 ≈ 0.943 × 1 0 5 Pa Yeh step kyun? Atmospheric se kam gas open side ko girne deta hai, isliye hum subtract karte hain.
Common mistake "Pressure tube ki shape/width par depend karta hai."Kyun sahi lagta hai: moti tube mein zyada fluid hota hai, zyada weight — zaroor zyada pressure hoga?
Fix: Pressure force per area hai. Zyada fluid zyada area par baitha hota hai, toh cancel ho jaata hai. Sirf vertical height h
Common mistake "Inclined tube meinh Kyun sahi lagta hai: fluid poori woh doori travel karta hai.
Fix: Sirf vertical rise gravity ke against weight contribute karta hai. h = L sin θ θ
Common mistake "Barometer absolute padhta hai, manometer bhi absolute padhta hai."Kyun sahi lagta hai: dono ρ g h Fix: Barometer ka top vacuum hai ⇒ yeh absolute P a t m P a t m gauge (P − P a t m P a t m
Δ P = ρ g h Same fluid + same level ⇒ same pressure.
Barometer: P a t m = ρ g h
Manometer: P g a s = P a t m ± ρ g h
Recall Feynman: 12-saal ke bachche ko samjhaoPool mein dive karne ka socho. Surface ke paas kaano ko theek lagta hai, lekin gehra jao toh dard hota hai — woh "dard" pressure hai, aur badhta hai kyunki tumhare upar zyada paani baitha hai. Ab ulta socho: agar humein pata ho kaano mein kitna dard hai, toh hum guess kar sakte hain tum kitne gehere ho! Barometer aur manometer bilkul yahi karte hain liquid ki patli tube se. Hawa (ya trapped gas) liquid par push karti hai aur use upar uthati hai; hum bas kitna ooncha liquid chadha yeh measure karte hain, aur woh height hi pressure hai. Mercury isliye use hota hai kyunki woh bahut bhaari hai, toh strong air pressure bhi use thoda hi uthata hai — tumhari desk par ek tube mein fit ho jaata hai.
"Deeper = Denser-push = Down adds, Up subtracts."
Aur devices ke liye: B arometer = B lank (vacuum) upar = aB solute; M anometer = M eets atmosphere = gauM ge (gauge). h vertical gap hota hai.
Depth h Δ P = ρ g h
Tube ki cross-sectional area pressure ko affect kyun nahi karti? Pressure = force/area; extra fluid weight proportionally zyada area par baitha hota hai, toh A
Mercury barometer tube ke top par kya hota hai? Ek vacuum (P ≈ 0 P a t m = ρ g h
Standard atmospheric pressure mmHg aur Pa mein? 760 mmHg ≈ 1.013 × 1 0 5
Barometer mein water ki jagah mercury kyun? Mercury 13.6 × h = P / ρ g
Open manometer: gas pressure formula? P g a s = P a t m ± ρ g h + −
Open-tube manometer gauge ya absolute pressure padhta hai? Gauge (ρ g h P a t m
θ L h h = L sin θ
Same connected static fluid mein same height par do points ki...? Same pressure hoti hai.
Gauge pressure ki definition? P g a ug e = P ab s − P a t m = ρ g h
Pascal's Law — pressure equally transmit hota hai; "same level = same pressure" ka base.Hydrostatic Pressure — parent ρ g h Buoyancy and Archimedes Principle — yeh bhi depth-dependent pressure par based hai.Atmospheric Pressure — jo barometer measure karta hai.Bernoulli's Equation — pressure ko moving fluids tak generalize karta hai.Density and Specific Gravity — manometric fluid ka choice (ρ
same level equal pressure
Pressure increases with depth
Force balance on cylinder
Hydrostatic law P2 = P1 + rho g h
Fluid column as measuring stick
Mercury chosen over water
