Hydrostatics — pressure = ρgh, derivation

2.2.5 · Physics › Fluid Mechanics

KISKE baare mein baat kar rahe hain?

KYUN zaroori hai? Dams, submarines, blood pressure, barometers, buoyancy — inme se har ek ka existence is baat pe depend karta hai ki depth ke saath pressure kaise badalti hai.

KAISE: $P = ρ g h$ ko scratch se derive karo

Hum sirf Newton ka first law use karenge: ek fluid jo rest mein hai, uske har piece par net force zero hoti hai.

Figure — Hydrostatics — pressure = ρgh, derivation

Step 1 — Body choose karo. Ek vertical cylinder of fluid: area $A$ , height $h$ , top free surface par (atmosphere ke liye open), bottom depth $h$ par.

Yeh step kyun? Ek column geometry ko simple banata hai — vertical balance ke liye sirf top aur bottom faces matter karte hain; sides horizontally push karte hain aur cancel ho jaate hain.

Step 2 — Column ki mass nikalo. $V = A h, m = ρ V = ρ A h$

Yeh step kyun? Pressure weight ke baare mein hai, aur weight ke liye mass chahiye; mass aati hai density × volume se.

Step 3 — Column par forces (sirf vertical).

Weight neeche kheench raha hai: $W = m g = ρ A h g$
Atmosphere neeche top par push kar raha hai: $F_{top} = P_{0} A$
Neeche ka fluid upar bottom par push kar raha hai: $F_{bot} = P A$ (yahi hume chahiye)

Yeh step kyun? Har face pressure $\times$ area feel karta hai. Curved wall par sideways forces horizontal hain aur symmetry se cancel ho jaate hain, isliye hum unhe ignore karte hain.

Step 4 — Equilibrium apply karo (net force $= 0$ ). $F_{bot} = F_{top} + W$ $P A = P_{0} A + ρ A h g$

Yeh step kyun? Column rest mein hai $\Rightarrow$ upar ka push balance karta hai (neeche ka push + weight).

Step 5 — $A$ se divide karo. $P = P_{0} + ρ g h$

Sirf fluid ki wajah se gauge (extra) pressure hai: $P_{gauge} = P - P_{0} = ρ g h$

Worked examples

Common mistakes (Steel-man + fix)

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho ek lamba stack of pillows hai. Agar tum stack ke bottom par lete ho, toh bahut saare pillows tumhe daba rahe hain. Top ke paas, sirf kuch hi daba rahe hain. Paani same hai: jitna gehra ghuso, utna zyada paani tumhare upar hai aur tumhe squeeze karta hai. Do guna gehra $\Rightarrow$ roughly do guna squeeze. Aur koi fark nahi padta ki pool bada hai ya chhota — sirf yeh matter karta hai ki tum kitne gehra gaye ho, kyunki sirf tumhare seedha sir ke upar wala paani hi tumhe press kar raha hai.

Flashcards

Hydrostatic pressure formula (gauge)

P = ρ g h

ρ g h

mein har symbol ka kya matlab hai?

ρ

= fluid density,

g

= gravity,

h

= surface ke neeche vertical depth

Absolute vs gauge pressure relation

P_{ab s} = P_{0} + ρ g h

; gauge

= ρ g h

Kya pressure container ki shape/area par depend karta hai?

Nahi — sirf vertical depth

h

par (hydrostatic paradox)

P = ρ g h

derive karne mein kaun sa starting principle use hua?

Fluid at rest

\Rightarrow

kisi bhi column par net force zero hai (force balance)

Side forces derivation se kyun nikal jaati hain?

Woh horizontal hain aur symmetry se cancel ho jaati hain; sirf top/bottom faces vertical force dete hain

Roughly kitna depth ka paani 1 atmosphere add karta hai?

About 10 m

Pressure ki SI unit

pascal,

Pa = N/m^{2}

1 atm par mercury barometer ki height

\approx 0.76 m

(760 mm Hg)

1 g/cm^{3}

ko SI mein convert karo

1000 kg/m^{3}

Depth ke saath pressure kyun badhta hai (ek line mein)?

Upar zyada fluid weight baithta hai, same area par spread hoke

Connections

Pascal's Law — enclosed fluid mein apply kiya hua pressure equally transmit hota hai
Atmospheric Pressure & Barometer — $ρ g h$ ke zariye $P_{0}$ measure karna
Buoyancy & Archimedes' Principle — net upward force depth ke saath pressure difference se aata hai
Manometers — pressure differences ko height differences ke roop mein padhna
Bernoulli's Equation — motion add karta hai; fluid static ho toh $ρ g h$ ban jaata hai
Density and Specific Gravity — $ρ$ supply karta hai