Buoyancy — Archimedes' principle, derivation from pressure difference

2.2.7 · Physics › Fluid Mechanics

Buoyancy exist kyun karta hai?

KYA claim kiya ja raha hai (Archimedes' principle):

HOW to derive it from pressure (first principles)

Ek rectangular block lo jiska horizontal cross-section area $A$ aur height $H$ hai. Uska top face depth $h_{1}$ par hai, bottom face depth $h_{2} = h_{1} + H$ par. Fluid ki density $ρ_{f}$ hai.

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

Step 1 — Har horizontal face par Pressure. $p_{top} = p_{0} + ρ_{f} g h_{1}, p_{bottom} = p_{0} + ρ_{f} g h_{2}$ Yeh step kyun? Hydrostatic pressure sirf depth par depend karta hai, isliye har flat face par ek uniform pressure hoga.

Step 2 — Force = pressure × area (vertical sides ki forces cancel ho jaati hain). $F_{down} = p_{top} A, F_{up} = p_{bottom} A$ Yeh step kyun? Side faces opposite-facing pairs mein aate hain, equal depth par, isliye unki horizontal forces cancel ho jaati hain. Sirf top aur bottom hi net vertical force dete hain.

Step 3 — Net upward force. $F_{B} = F_{up} - F_{down} = (p_{bottom} - p_{top}) A$ $= [(p_{0} + ρ_{f} g h_{2}) - (p_{0} + ρ_{f} g h_{1})] A = ρ_{f} g (h_{2} - h_{1}) A$ Yeh step kyun? $p_{0}$ atmospheric terms cancel ho jaate hain — buoyancy purely depth difference se aati hai.

Step 4 — Volume ko pehchano. $h_{2} - h_{1} = H \Rightarrow F_{B} = ρ_{f} g (H A) = ρ_{f} g V$ Yeh step kyun? $H A$ bilkul block ka volume hai = us fluid ka volume jo usne displace kiya.

Floating vs sinking (principle apply karo)

Ek object ki density $ρ_{o}$ , volume $V$ , weight $W = ρ_{o} V g$ ke liye:

Sinks agar $F_{B} < W \Rightarrow ρ_{f} < ρ_{o}$
Floats (upar uthta hai) agar $F_{B} > W \Rightarrow ρ_{f} > ρ_{o}$
Floating in equilibrium partly submerged: sirf fraction $f$ paani ke andar hai, aur $ρ_{f} (f V) g = ρ_{o} V g \Rightarrow f = \frac{ρ _{o}}{ρ _{f}}$

Worked Examples

Common Mistakes (Steel-manned)

Recall Feynman: 12-saal ke bachche ko explain karo

Pool ke neeche paani upar wale paani ke wajan se dabaa hua hai, isliye woh upar wale paani se zyada force se push karta hai. Jab tum ek ball neeche daalte ho, neeche se zyada push upar ki halki push ko beat kar deta hai, isliye ball upar dhakeli jaati hai. Yeh dhakka bilkul us paani ke wajan ke barabar hota hai jo ball ne apni jagah se hataya. Agar ball us paani se halki hai, toh woh upar aa jaati hai; agar bhaari hai, toh neeche chali jaati hai.

Active Recall

Archimedes' principle kya hai?

Ek immersed object ko ek upar ki taraf force feel hoti hai jo us fluid ke weight ke barabar hai jo usne displace kiya,

F_{B} = ρ_{f} V_{d i s p} g

Buoyancy physically kahan se aati hai?

Bottom face par pressure top se zyada hota hai (gehra fluid = zyada pressure); net upward pressure force hi buoyancy hai.

Buoyancy derivation mein atmospheric pressure

p_{0}

kyun cancel ho jaata hai?

Yeh top aur bottom dono faces par equally act karta hai, isliye subtract ho jaata hai; sirf depth-difference term

ρ_{f} g H

bachta hai.

F_{B}

mein kaunsi density appear karti hai, object ki ya fluid ki?

Fluid ki density

ρ_{f}

(fluid hi push kar raha hai).

Kya buoyant force object ki depth par depend karta hai?

Nahi (incompressible fluid ke liye, fully submerged); yeh displaced volume par depend karta hai

p_{b o tt o m} - p_{t o p} = ρ_{f} g H

ke zariye.

Object float karne ki condition kya hai?

ρ_{o bj ec t} < ρ_{f l u i d}

(tab full submersion par

F_{B}

weight se zyada ho jaata hai).

Floating body ka submerged fraction?

f = ρ_{o bj ec t} / ρ_{f l u i d}

Submerged body ka apparent weight?

W_{a pp a r e n t} = W_{ai r} - F_{B}

Steel ka ship kyun float karta hai?

Uska hull hawa enclose karta hai isliye average density paani se kam ho jaati hai, itna paani displace ho jaata hai ki uske weight ko balance kar sake.

Buoyant force kahan act karta hai?

Centre of buoyancy par = displaced fluid volume ka centroid.

Connections

Pressure in fluids — hydrostatic pressure ( $p = p_{0} + ρ g h$ ka source)
Density and relative density
Floating bodies and stability — metacentre
Pascal's principle
Newton's laws — equilibrium of forces
Apparent weight and weighing methods