2.2.8 · D3Manometers, barometers
Intuition Ye page kis kaam ki haiParent note ne tumhe do master tools diye the:
Δ P = ρ g h neeche jaane par vertical height h ρ g h har tarah ke problem fire karta hai jo un tools ke saamne aa sakte hain — har sign, har degenerate case, har twist — aur har ek ko line one se solve karta hai.
Agar tumne abhi tak ρ g h Hydrostatic Pressure padho.
Kuch bhi shuru karne se pehle, un letters ko pin down karte hain jo baar baar use honge, plain words mein:
Definition Symbols, anchored
ρ density = ek cubic metre mein kitne kilograms fluid pack hain. Paani 1000 kg/m 3 13600 kg/m 3 13.6 × Density and Specific Gravity .g = 9.8 m/s 2 h vertical height difference — seedha upar-neeche , kabhi bhi tilted distance nahi.P a t m atmospheric pressure = aas-paas ki hawa jo kisi bhi exposed surface par dabaati hai. Sea level par ye roughly 1.013 × 1 0 5 Pa Atmospheric Pressure .
Pehle teen ko multiply karo, ρ g h Pa pressure milega. Units check: m 3 kg ⋅ s 2 m ⋅ m = m ⋅ s 2 kg = m 2 N = Pa
Do aur words jin par hum constantly lean karenge — inhe abhi meet karo taaki koi example surprise na kare:
Definition Absolute vs gauge pressure
Absolute pressure P ab s total, sachchi pressure, sab kuch count karke (atmosphere jo neeche push karta hai usse bhi). Zero absolute matlab perfect vacuum.Gauge pressure P g a ug e P g a ug e = P ab s − P a t m = ρ g h Dono ki zaroorat kyun? Zyaatar instruments atmosphere mein baithe hote hain aur sirf difference feel karte hain, isliye naturally gauge report karte hain. Absolute total paane ke liye, P a t m
Definition Reference line aurP l in e reference line kehte hain. Us line par pressure ke liye hum P l in e P l in e same value hoti hai chahe hum ise left arm se walk karke compute karein ya right arm se — aur un dono computations ko equal karna hi manometers solve karne ka poora trick hai.
Har manometer/barometer problem in cells mein se ek mein rehta hai. Neeche ke examples har ek cell fill karte hain taaki koi gap na rahe.
#
Cell (kya alag hai)
Kaun sa example
A
Barometer, vacuum on top → absolute P a t m
Ex 1
B
Barometer with non-mercury fluid (limiting height)
Ex 2
C
Barometer with trapped gas (perfect vacuum nahi)
Ex 3
D
Manometer, open arm higher → gas above atmospheric (+ sign)
Ex 4
E
Manometer, gas arm higher → gas below atmospheric (− sign)
Ex 5
F
Degenerate : levels equal, h = 0 Ex 6
G
Inclined manometer → h = L sin θ Ex 7
H
Do immiscible fluids stacked → har ρ g h Ex 8
I
Real-world word problem (blood-pressure / diver style)Ex 9
J
Exam twist : g Ex 10
Recall Do rules jo neeche sab kuch use karta haiRule 1 — fluid path pe chalo: neeche jaane par ρ g h add hota hai, upar jaane par ρ g h subtract hota hai.
Rule 2 — same connected fluid, same horizontal level ⇒ equal pressure .
Worked example Ex 1 — Standard mercury barometerEk mercury barometer column height h = 0.760 m P a t m ρ H g = 13600 kg/m 3 g = 9.8
Forecast: aage padhne se pehle pressure guess karo — kya ye 1 0 5 Pa
Same level par do points choose karo: point A, tube ke base ke andar mercury surface par, point B, open dish surface par. Figure s01 dekho — A (mint dot) aur B (butter dot) same dashed level par baithe hain.
Ye step kyun? Rule 2 inhe tabhi equate karne deta hai jab ye ek horizontal level share karte hain.B (open) par pressure likho: P B = P a t m Kyun? B hawa ke samne exposed hai, isliye hawa ka pressure directly uus par baithta hai.A par pressure likho: A se upar vacuum top tak column climb karo (figure mein coral label). Upar jaane par subtract hota hai, isliye top pressure = P A − ρ g h = 0 P A = ρ g h Kyun? Vertical tube mein Rule 1; top empty hai ⇒ zero.P A = P B ρ g h = P a t m Inhe equate kyun kar sakte hain? Rule 2 se: A (tube ke base ke andar) aur B (open dish surface) same connected mercury mein same horizontal level par hain, isliye unhe ek pressure share karna hi hoga — warna fluid sideways flow karta high-pressure point se tab tak jab tak match na ho jaayein.
P a t m = 13600 × 9.8 × 0.760 = 1.013 × 1 0 5 Pa
Verify: 760 mmHg = 1 atm 1 atm = 1.013 × 1 0 5 Pa m 3 kg ⋅ s 2 m ⋅ m = Pa
Upar wali figure exactly ye dikhati hai: coral double-arrow column height h
Worked example Ex 2 — Water barometer heightAgar hum mercury ki jagah water (ρ = 1000 P a t m = 1.013 × 1 0 5 Pa
Forecast: mercury ko 0.76 m 13.6 ×
Shuru karo P a t m = ρ g h h h = ρ g P a t m Ye step kyun? Pressure pata hai aur height chahiye, isliye rearrange karo.Water ki density daalo:
h = 1000 × 9.8 1.013 × 1 0 5 = 10.34 m
Verify: ratio h H g h w a t er = ρ w a t er ρ H g = 1000 13600 = 13.6 0.760 × 13.6 = 10.34 m Atmospheric Pressure .
Figure s02 dono columns ko scale ke saath side by side rakhti hai: chhota lavender mercury bar (0.76 m 10.34 m same P a t m
Worked example Ex 3 — Imperfect vacuumEk mercury barometer ke top par thodi si trapped air hai jo P t o p = 2000 Pa h = 0.745 m P a t m
Forecast: trapped gas mercury ko neeche push karta hai, isliye column low read karta hai. Kya sachcha P a t m ρ g h
Same-level trick reuse karo: bilkul Figure s01 ki tarah, dish surface aur in-tube surface ko ek level par lo, phir tube climb karo; lekin ab top P t o p 0 Ye step kyun? Rule 2 abhi bhi guarantee karta hai ki same-level points ek pressure share karte hain; Ex 1 se ek hi cheez badli hai ki column ki roof ab ek real gas hai, vacuum nahi, isliye uska pressure climb mein enter hota hai.Balance: P a t m = P t o p + ρ g h Ye step kyun? Column upar jaate waqt ρ g h P t o p P a t m = P t o p + ρ g h P a t m = 2000 + 13600 × 9.8 × 0.745 = 2000 + 99287 = 1.013 × 1 0 5 Pa
Verify: agar P t o p = 0 h = 0.760 0.745 same P a t m
Worked example Ex 4 — Pressurised gas (+ sign)Ek open-tube mercury manometer ek gas vessel se connected hai. Open arm liquid h = 0.05 m higher hai gas-arm liquid se. P a t m = 1.01 × 1 0 5 Pa
Forecast: agar gas open side ko upar push kar sakta hai, toh gas air se stronger hai ya weaker?
Reference line = lower liquid surface (gas side). Figure s03 mein ye mint dashed line hai; P l in e Right (open) arm, uski surface se line tak neeche: P l in e = P a t m + ρ g h Kyun? Open surface se h neeche jaane par (figure mein coral arrow) ρ g h Left (gas) arm, gas se line tak neeche: P l in e = P g a s hi line hai).P l in e P g a s = P a t m + ρ g h P ab s = 1.01 × 1 0 5 + 13600 × 9.8 × 0.05 = 1.01 × 1 0 5 + 6664 = 1.077 × 1 0 5 Pa
Gauge: P g a ug e = P ab s − P a t m = 6664 Pa
Verify: open arm higher ⇒ gas air ko overcome karta hai ⇒ P g a s > P a t m +
Figure s03 mein right (open) liquid, left (gas) liquid se uuncha khada hai; coral arrow height gap h P l in e upar hai.
Worked example Ex 5 — Partial vacuum (− sign)Same manometer, lekin ab gas arm liquid 0.05 m higher hai (atmosphere ne open side ko neeche push kar diya). P a t m = 1.01 × 1 0 5
Forecast: is tug-of-war mein hawa jeeti. Gas atmospheric se upar hai ya neeche?
Reference line = lower surface (ab open side); P l in e Left (gas) arm line tak neeche: P l in e = P g a s + ρ g h Right (open) arm line tak neeche: P l in e = P a t m P l in e P g a s + ρ g h = P a t m ⇒ P g a s = P a t m − ρ g h P ab s = 1.01 × 1 0 5 − 6664 = 0.943 × 1 0 5 Pa
Gauge: − 6664 Pa
Verify: gas arm higher ⇒ gas air se kamzor ⇒ P g a s < P a t m −
h = 0 exactly same height par hain. Gas pressure kya hai?
Forecast: kuch bhi kuch bhi upar ya neeche nahi push kar raha. Prediction?
Apply karo P g a s = P a t m ± ρ g h h = 0 P g a s = P a t m
Verify: gauge pressure = ρ g ⋅ 0 = 0 + −
Worked example Ex 7 — Slanted tubeEk manometer ka open arm ek tube hai jo horizontal se θ = 3 0 ∘ slant length L = 0.10 m ρ = 850 kg/m 3
Forecast: hum 0.10 m sabhi height count hota hai?
Vertical rise nikalo. Sirf up-down part gravity se ladhta hai, isliye h = L sin θ L h sin cos θ θ L sin θ L cos θ h = 0.10 × sin 3 0 ∘ = 0.10 × 0.5 = 0.05 m Ab use karo P g a ug e = ρ g h P g a ug e = 850 × 9.8 × 0.05 = 416.5 Pa
Verify: agar hum galti se slant L = 0.10 833 Pa lamba, easy-to-read slant pe spread kar dein. ✓
Figure s04 h = L sin θ sin
Worked example Ex 8 — Water on top of mercuryEk U-tube mein neeche mercury hai aur left arm mein mercury ke upar 0.20 m water (ρ w = 1000 h H g = 0.014 m P X
Forecast: do alag fluids ⇒ do alag ρ g h
Reference line = lower mercury surface (left arm base); P l in e Left arm, top se line tak neeche: P X 0.20 m neeche jaao, phir line tak neeche. Isliye P l in e = P X + ρ w g ( 0.20 ) Split kyun karo? Har fluid apna khud ka ρ g h Right (open) arm line tak neeche: P l in e = P a t m + ρ H g g h H g P l in e P l in e P X + ρ w g ( 0.20 ) = P a t m + ρ H g g ( 0.014 ) . P X ρ w g ( 0.20 ) P X = P a t m + ρ H g g ( 0.014 ) − ρ w g ( 0.20 ) . Numbers daalo: ρ H g g ( 0.014 ) = 13600 ( 9.8 ) ( 0.014 ) = 1866.6 Pa ρ w g ( 0.20 ) = 1000 ( 9.8 ) ( 0.20 ) = 1960 Pa P X = P a t m + 1866.6 − 1960 = P a t m − 93.4 Pa .
Verify (units & sign): har term ρ g h P X alag alag add karna hi key skill hai. ✓
Worked example Ex 9 — Diver ka kaanEk swimmer fresh water (ρ = 1000 h = 3.0 m Pa P a t m = 1.013 × 1 0 5
Forecast: parent ka Feynman story — gehra jaane par kaana hurt karta hai. Guess karo: 3 m par ek atmosphere se zyada hoga ya kam?
Depth par gauge pressure sirf ρ g h = 0 Δ P = 1000 × 9.8 × 3.0 = 29400 Pa Ye step kyun? Paani ki surface atmospheric par hai; har metre neeche jaane par ρ g h extra exactly gauge pressure hai.Atmosphere ke fraction ke roop mein: 1.013 × 1 0 5 29400 = 0.29
Verify: sirf 3 m paani se ≈ 29% Buoyancy and Archimedes Principle se bhi match karta hai, jo isi depth-pressure growth par rely karta hai. ✓
Worked example Ex 10 — Mars par BarometerEx 1 ka same mercury barometer ek aise planet par le jaaya jaata hai jahan gravity g ′ = 3.7 m/s 2 1.013 × 1 0 5 Pa
Forecast: kamzor gravity ⇒ mercury kam weighs karta hai ⇒ same air pressure balance karne ke liye column zyada uuncha jaayega ya neeche baithega?
P a t m = ρ g ′ h h = ρ g ′ P a t m Ye step kyun? Sirf g ρ P a t m h h = 13600 × 3.7 1.013 × 1 0 5 = 2.01 m
Verify: ratio h E a r t h h ′ = g ′ g E a r t h = 3.7 9.8 = 2.65 0.760 × 2.65 = 2.01 m
Common mistake Teen sign traps jo ye page defeat karta hai
Manometer sign: + ρ g h open arm higher ho (gas strong hai); − ρ g h gas arm higher ho (gas weak hai). Ex 4 vs Ex 5.Slant vs vertical: hamesha h = L sin θ Mixed fluids: har fluid apna khud ka ρ g h
Mnemonic Ek line jo sab carry kare"Neeche jaao add karo, upar jaao subtract karo; open-higher plus, gas-higher minus; sirf vertical count hoti hai."
