Hydrostatics — pressure = ρgh, derivation: Foundations

2.2.5 · D1 · Physics › Fluid Mechanics › Hydrostatics — pressure = ρgh, derivation

Is page par kuch bhi assume nahi kiya gaya. Parent derivation padhne se pehle, tumhe aath chhoti ideas ke saath fluent hona chahiye. Hum inhe ek-ek karke build karte hain, har ek pichle ke upar, har ek ek picture se anchored. Agar kuch pehle se pata hai, skim kar lo — lekin parent jo bhi symbol use karta hai, sab yahaan hai.

1. Force — ek dhakka ya kheench

Topic ko yeh kyun chahiye. Ek fluid apne container ki walls aur floor par press karta hai. Woh press hi ek force hai. Agar hum pushes aur pulls ke baare mein baat nahi kar sakte, toh pressure ke baare mein ek bhi baat nahi kar sakte.

Ek khaas force baar-baar aati hai: weight. Weight kisi object ki mass par gravity ka neeche ki taraf pull hai. Hum ise step 5 mein theek se build karenge jab mass aur gravity mil jayenge.

2. Area — ek flat surface ka size

Topic ko yeh kyun chahiye. Wahi push ek bade surface par spread ho toh gentle lagti hai; wahi push ek chote surface par concentrate ho toh sharp lagti hai. Pressure bilkul yahi "push per surface" hai, isliye hume surface measure karna aana chahiye. Derivation mein, fluid column ke top aur bottom dono ka area $A$ hota hai.

Figure — Hydrostatics — pressure = ρgh, derivation

Amber arrow (force) aur cyan patch (area) dekho. Wahi arrow, do patch sizes — chote patch ko crush hota hai, bade patch ko mushkil se kuch lagta hai. Yahi hai "per area" ka poora matlab.

3. Pressure — force spread over area

Humne force ÷ area choose kiya (force ÷ kuch aur nahi) kyunki paani ke andar hum jo actually feel karte hain woh poore pool floor par total push nahi hai — woh hai ki push hamare apne chote skin patch par kitni concentrated hai.

Hum baad mein is relation ko reverse karenge: agar hume pata hai ki pressure $P$ area $A$ ke face par act kar raha hai, toh us face par force hai $F = P \times A$ . Parent mein, iska use har pressure ko column ke top aur bottom faces par force mein badalne ke liye hota hai.

4. Mass, volume, aur density — kitna "stuff" hai

Teen ideas saath chalti hain. Humhe inki zaroorat hai kyunki pressure weight se aata hai, aur weight ke liye mass chahiye, aur fluid ki mass uski density times us ka volume se aati hai.

Topic ko yeh kyun chahiye. Derivation mein hum fluid ka ek imaginary column cut karte hain. Uska weight nikalne ke liye pehle uski mass chahiye, aur constant density se us column ki mass simply $m = ρ V = ρ (A h)$ hai.

5. Gravity $g$ — cheezein weight kyun rakhti hain

Topic ko yeh kyun chahiye. Depth ke saath pressure exist karta hai sirf isliye kyunki fluid bhari hai. Hamare tools combine karke — $W = m g$ , phir $m = ρ V$ , phir $V = A h$ — ek fluid column ka weight hai $W = m g = (ρ V) g = ρ (A h) g,$ jo poori derivation ka heart hai.

6. Depth $h$ — seedha neeche measure kiya

7. Top par pressure $P_{0}$ — head start

8. Equilibrium — balance rule jo derivation chalata hai

Figure column par teen vertical arrows dikhata hai. Do sideways cyan arrows wall par notice karo: woh horizontal hain, isliye left-against-right cancel hoti hain aur vertical balance mein kabhi nahi aatein. Yahi wajah hai ki sirf top aur bottom faces matter karte hain.

Yeh foundations topic ko kaise feed karte hain

Upar se padho: force aur area pressure banate hain; area aur depth volume banate hain, jo constant density ke saath mass banata hai, jo gravity ke saath weight banata hai; equilibrium pressure aur weight ko final formula mein jodata hai.

Equipment checklist

Apne aap ko test karo — derivation ke liye ready ho sirf tab jab har reveal woh ho jo tum kehte.

Pressure kya hai, words aur symbols mein?

Force spread over area,

P = F / A

, pascals (

N/m^{2}

) mein measure hota hai

Agar area

A

ke face par pressure

P

act kar raha hai, toh woh kitni force produce karta hai?

F = P \times A

F_{top}

aur

F_{bot}

mein subscripts ka matlab kya hai?

Sirf naam hain — column ke top face aur bottom face par force

Area

A

aur height

h

ke seedhe column ka volume kya hai?

V = A h

Density kya hai aur uska formula?

Mass per unit volume,

ρ = m / V

kg/m^{3}

mein

Fluid ke ek chunk ki mass density se kaise nikalte hain?

m = ρ V = ρ A h

Derivation

ρ

ke baare mein kya key assumption karta hai?

Constant density (incompressible) — har depth par same

ρ

Weight kya hai aur yeh mass se kaise alag hai?

Weight

= m g

newtons mein ek downward force hai; mass kg mein stuff ki quantity hai

g

ki value aur matlab?

g \approx 9.8 m/s^{2}

, gravity ka pull per kilogram

Depth

h

kaise measure hoti hai?

Vertically, free surface se seedha neeche — kabhi tilted ya horizontal nahi

P_{0}

kya hai?

Woh pressure jo top free surface par neeche push kar raha hai, usually atmospheric

\approx 1.01 \times 1 0^{5} Pa

Ruke hue fluid ke liye equilibrium rule batao.

Zero net force: saari up-forces saari down-forces ke barabar hoti hain

Column par side forces kyun drop out ho jaati hain?

Woh horizontal hain aur symmetry se cancel hoti hain; sirf top aur bottom vertical force dete hain

Absolute aur gauge pressure mein kya fark hai?

Absolute

= P_{0} + ρ g h

(total); gauge

= ρ g h

(

P_{0}

se upar extra)

1 g/cm^{3}

ko SI mein convert karo.

1000 kg/m^{3}

Connections

Parent topic — poori $P=\rho g h$ derivation
Density and Specific Gravity — $ρ$ supply karta hai
Atmospheric Pressure & Barometer — $P_{0}$ supply karta hai
Pascal's Law — gauge pressure kya transmit karta hai
Manometers — gauge pressure ko height ke roop mein padhna
Buoyancy & Archimedes' Principle — depth ke saath pressure difference use karta hai
Bernoulli's Equation — is static picture mein motion add karta hai