Buoyancy — Archimedes' principle, derivation from pressure difference: Foundations

2.2.7 · D1 · Physics › Fluid Mechanics › Buoyancy — Archimedes' principle, derivation from pressure d

Parent note Buoyancy mein Archimedes' principle derive karne se pehle, humein us mein aane waale har letter ka poora knowledge hona chahiye. Yeh page har ek cheez ko bilkul scratch se banata hai — pehle simple words mein, phir ek picture, phir yeh reason ki yeh topic uske bina exist nahi kar sakta.

0. Fluid aakhir hota kya hai?

Neeche diye figure ko dekho. Chote chote arrows fluid ko bahar ki taraf push karte hue dikhate hain — tank ki walls pe aur ek dube hue object pe. Notice karo ki arrows hamesha us surface ke right angles par hote hain jise woh hit karte hain — "fluid pressure" aisa hi dikhta hai.

Figure — Buoyancy — Archimedes' principle, derivation from pressure difference

Yeh topic ko kyun chahiye: poora topic upar aur neeche se aane wali pushes ke beech ki ladai hai. Toh hume pehle yeh agree karna hoga ki fluid push karta hai, aur hamesha surface ke perpendicular.

1. Depth $h$ — tum kitne neeche ho

Picture mein, block ka top face depth $h_{1}$ par hai aur bottom face aur neeche, depth $h_{2}$ par hai. $h$ label wala arrow hamesha seedha neeche ki taraf point karta hai, kabhi sideways nahi.

Yeh topic ko kyun chahiye: pressure depth par depend karta hai. Do faces jo alag alag depths par hain unhe alag alag push feel hoti hai — yahi difference buoyancy ka beej hai.

2. Density $ρ$ — cheez kitni tightly packed hai

Do identical boxes imagine karo: ek mein bahut saari choti balls bhari hain (high $ρ$ ), ek mein thodi si (low $ρ$ ). Same box, alag mass — yahi density hai.

Is topic mein hum do densities se milte hain, aur unhe mix up karna classic galti hai:

$ρ_{f}$ — fluid ki density (jo paani/hawa push kar raha hai).
$ρ_{o}$ — object ki density (jo cheez push ho rahi hai).

Yeh topic ko kyun chahiye: pressure fluid ki density ke saath badhta hai, aur float/sink karna $ρ_{o}$ aur $ρ_{f}$ ko compare karke decide hota hai.

3. Gravity $g$ — woh pull jo fluid ko stack karta hai

Gravity isliye hai ki gehra fluid squeeze hota hai: fluid ki har layer ka weight hota hai, aur layers ek doosre ke upar pile hoti jaati hain. Upar zyada fluid stacked ⇒ neeche zyada squeeze.

Yeh topic ko kyun chahiye: gravity ke bina koi weight nahi, koi stacking nahi, depth ke saath koi pressure nahi, aur isliye buoyancy bhi nahi. Yahan har formula mein $g$ hoga.

4. Pressure $p$ — area par spread hua push

Drawing pin press karne ke baare mein socho. Same push, lekin ek tiny point par concentrated (small $A$ ) matlab huge pressure. Poori palm par spread karo (big $A$ ), mushkil se feel hota hai. Pressure kitna concentrated hai push, yahi batata hai.

Neeche wala figure ek block ke flat bottom face par pressure ko dikhata hai: uniform chote arrows area $A$ par ek bade force $F = p A$ mein add ho jaate hain.

Yeh topic ko kyun chahiye: fluid humein faces par pressures deta hai, lekin Newton's laws ko forces chahiye. Yeh equation woh bridge hai, jo parent derivation ke Step 2 mein use hoti hai.

5. Hydrostatic pressure — depth formula

Ab depth (§1), density (§2), gravity (§3) ko buoyancy ke peeche ke sabse important formula mein combine karo.

Figure do faces dikhata hai block ki depths $h_{1}$ (top) aur $h_{2}$ (bottom) par: deeper face ek taller fluid column ke neeche baitha hai, isliye ise larger pressure feel hota hai. Yeh ek nazar mein buoyancy ki poori origin hai.

Poori story ke liye dekho Pressure in fluids — hydrostatic pressure.

Yeh topic ko kyun chahiye: yahi engine hai. $p_{bottom} - p_{top} = ρ_{f} g (h_{2} - h_{1})$ literally buoyant force per unit area hai.

6. Volume $V$ aur displaced volume $V_{disp}$

Key picture: poora duba hua object apna poora volume displace karta hai. Aadha floating object sirf paani ke andar wala hissa displace karta hai.

Yeh topic ko kyun chahiye: Archimedes' result $F_{B} = ρ_{f} V_{disp} g$ simply yeh hai ki "us fluid ka weight jo pehle us hole mein tha." Volume hi density ko mass mein, aur mass ko weight mein convert karta hai.

7. Net force & equilibrium — push fight mein kaun jeeta

Buoyancy ke liye hum ek floating object par teen vertical forces line up karte hain: weight $W$ neeche pull kar raha hai, buoyancy $F_{B}$ upar push kar raha hai, aur (agar scale se hang kiya gaya ho) tension. Balance ka matlab hai $F_{B} = W$ . Yeh pure Newton's laws — equilibrium of forces hai.

Yeh topic ko kyun chahiye: "floats vs sinks vs aadha float" entirely up-push $F_{B}$ aur down-pull $W$ ko compare karke decide hota hai. Aur Apparent weight and weighing methods exactly yahi use karta hai: ek scale $W - F_{B}$ read karta hai.

Prerequisite map

Isko upar se neeche padho: gravity + density + depth se hydrostatic pressure banta hai; deeper face pe pressure top face ko beat karta hai; usse force mein convert karke add karo toh net upthrust milta hai; volume laao toh Archimedes' principle milta hai, jo phir floating decide karta hai.

Equipment checklist

Khud test karo — right side cover karo aur reveal karne se pehle answer do.

Fluid har surface ke saath kya karta hai jise woh touch karta hai, aur kis direction mein?

Woh surface ko push karta hai, hamesha us par perpendicular (right angles par).

Depth

h

kya hai aur ise kis direction mein measure kiya jaata hai?

Fluid ki top surface se seedha neeche ki taraf measure ki gayi vertical distance.

Density

ρ

ko uske formula aur units ke saath define karo.

Mass per unit volume,

ρ = m / V

kg/m^{3}

mein.

ρ_{f}

aur

ρ_{o}

mein kya fark hai?

ρ_{f}

fluid ki density hai (push karta hai);

ρ_{o}

object ki density hai (float/sink decide karta hai).

g

kya represent karta hai aur uski value kya hai?

Gravitational field strength / pull per kilogram, lagbhag

9.8 N/kg

Force, pressure aur area ko link karne wali equation likho.

F = p A

(pressure ko us area se multiply karo jis par woh act karta hai).

Hydrostatic pressure formula likho aur har term ka naam batao.

p = p_{0} + ρ_{f} g h

;

p_{0}

surface pressure,

ρ_{f} g h

upar fluid column ka extra push.

ρ_{f} g h

column ki width par kyun depend nahi karta?

Area

A

cancel ho jaata hai: weight

ρ_{f} A h g

ko area

A

se divide karo toh

ρ_{f} g h

bachta hai.

Buoyancy derivation mein

p_{0}

kyun cancel ho jaayega?

Woh top aur bottom dono faces par equally press karta hai, isliye difference lete waqt woh subtract ho jaata hai.

Poori tarah dube hue object ke liye displaced volume kya hai?

Uska poora volume

V

Kaunsi condition ka matlab hai ki object equilibrium mein hai (float karta hai)?

Upar wali buoyant force neeche wale weight ke barabar hoti hai,

F_{B} = W

; net force zero hai.

Connections

Pressure in fluids — hydrostatic pressure — jahan $p = p_{0} + ρ g h$ build hota hai
Density and relative density — $ρ$ ka meaning
Newton's laws — equilibrium of forces — $F_{B}$ aur $W$ balance karna
Apparent weight and weighing methods — ek scale $W - F_{B}$ read karta hai
Floating bodies and stability — metacentre — jahan buoyancy act karta hai
Pascal's principle — enclosed fluids mein pressure