Pascal's law — pressure transmits equally — Hinglish notes

2.2.6 · Physics › Fluid Mechanics

Pascal's Law KYA hai?

Is ek sentence ke andar do ideas chhupe hain:

Ek level par har jagah same pressure — ek height par static fluid mein, pressure har direction mein aur us level ke har point par equal hota hai.
Changes poori tarah transmit hote hain — agar ek point par pressure $Δ p$ badhao, toh har point par wahi $Δ p$ badhta hai.

Yeh sach KYU hai? (First principles se derivation)

Setup: Ek chote se wedge-shaped fluid element (triangular prism) lo thickness ke saath, ek static fluid ke andar. Teen faces par pressures ho: $p_{x}$ (vertical face par), $p_{y}$ (horizontal face par), $p_{n}$ (slanted face par). Element itna chota hai ki hum gravity ignore kar sakte hain (volume $\to 0$ , area se zyada tezi se).

Maan lo slanted face angle $θ$ banata hai, areas ke saath:

vertical face area $= d A_{x}$
horizontal face area $= d A_{y}$
slant face area $= d A_{n}$

Wedge ki geometry se milta hai: $d A_{x} = d A_{n} sin θ, d A_{y} = d A_{n} cos θ$

Horizontal balance (x): vertical face par pressure force = slant face par force ka horizontal component $p_{x} d A_{x} = p_{n} d A_{n} sin θ$

Yeh step kyun? Pressure ek surface ke andar perpendicular push karta hai; sirf slant-face force ka horizontal component, vertical-face force ko oppose karta hai.

$d A_{x} = d A_{n} sin θ$ substitute karo: $p_{x} d A_{n} sin θ = p_{n} d A_{n} sin θ \Rightarrow p_{x} = p_{n}$

Vertical balance (y): isi tarah $p_{y} d A_{y} = p_{n} d A_{n} cos θ \Rightarrow p_{y} = p_{n}$

Isliye: $p_{x} = p_{y} = p_{n}$

Ab transmission wala part. Hydrostatic relation yaad karo (height $h$ ke fluid column se): $p = p_{0} + ρ g h$

Agar hum external/applied pressure $p_{0} \to p_{0} + Δ p$ badhate hain, toh har depth $h$ par: $p_{new} = (p_{0} + Δ p) + ρ g h = p_{old} + Δ p$

$ρ g h$ term unchanged rehta hai (same fluid, same heights), isliye increase $Δ p$ har point par identical hota hai. Yahi Pascal's law hai.

Figure — Pascal's law — pressure transmits equally

Ise USE kaise karein — worked examples

Common mistakes (steel-manned)

Flashcards

Pascal's law ek sentence mein

Ek enclosed incompressible fluid par apply kiya gaya pressure change, fluid ke har hisse aur container walls tak bina kisi kami ke transmit hota hai.

Ek static fluid mein kisi point par pressure har direction mein equal kyun hota hai?

Kyunki equilibrium mein ek tiny wedge element par net force zero hota hai; wedge geometry force karta hai ki

p_{x} = p_{y} = p_{n}

Hydraulic press force relation

F_{2} = F_{1} A_{2} / A_{1}

(kyunki

p_{1} = p_{2}

Hydraulic lift mein chhoti si force bada load kyun lift karti hai?

Wahi pressure ek bade area par act karta hai; force = pressure × area, aur

A_{2} / A_{1}

bada hota hai.

Kya hydraulic press free energy deta hai?

Nahi —

F_{1} d_{1} = F_{2} d_{2}

; tum distance ko force ke saath trade karte ho.

Agar tum surface pressure

Δ p

badhate ho, toh depth

h

par pressure kitna badhega?

Exactly

Δ p

se (

ρ g h

term unchanged rehta hai).

Pascal's law ke liye do conditions

Fluid enclosed (confined) aur incompressible hona chahiye.

1 cm aur 10 cm radii wale pistons ke liye mechanical advantage?

(10/1)^{2} = 100

Recall Feynman: 12-saal ke bacche ko samjhao

Socho ek balloon poori tarah paani se bhara hai, bina kisi haawa ke. Agar tum ek taraf apni ungli se poke karo, toh poora balloon har jagah se bahar aata hai — sirf wahan se nahi jahan tumne dabaya. Paani squash nahi ho sakta, isliye tumhara push kahi na kahi jaana chahiye, aur woh paani ke rubber ke har bit ko equally push karta hai. Car lift mein yahi trick use hoti hai: tum paani ki ek patli straw ko dabate ho, aur wahi push ek mote platform par phail jaata hai. Ek choti jagah par chhoti push ek badi jagah par giant push ban jaati hai — lekin tumhein apni patli straw ko bahut loooong door push karna padta hai car ko thoda sa lift karne ke liye. Kuch bhi free nahi; tum sirf "kitna zyada" aur "kitna door" ko swap karte ho.

Connections

Hydrostatic pressure — p = p0 + ρgh — woh $ρ g h$ term deta hai jise humne constant rakha.
Pressure — force per unit area — basic definition $p = F / A$ jo press mein use hoti hai.
Hydraulic systems — brakes, lifts, jacks — direct engineering application.
Archimedes' principle — buoyancy — isotropic fluid pressure par bhi rely karta hai.
Conservation of energy in machines — explain karta hai kyun press free-energy device nahi hai.
Incompressibility & continuity equation — kyun $A_{1} d_{1} = A_{2} d_{2}$ .

Pascal's Law KYA hai?

Yeh sach KYU hai? (First principles se derivation)

Ise USE kaise karein — worked examples

Common mistakes (steel-manned)

Flashcards

Connections

Concept Map