Hydrostatics — pressure = ρgh, derivation: Question bank

2.2.5 · D5 · Physics › Fluid Mechanics › Hydrostatics — pressure = ρgh, derivation

Shuru karne se pehle, symbols ka ek reminder taaki kuch surprise na ho:

$P$ = pressure (force per area, pascals mein $Pa = N/m^{2}$ ).
$ρ$ = fluid density (mass per volume, $kg/m^{3}$ ), dekho Density and Specific Gravity.
$g$ = gravitational field strength ( $\approx 9.8 m/s^{2}$ ).
$h$ = vertical depth below the free surface.
$P_{0}$ = surface par pressure (usually atmospheric).

True or false — justify

Har item ek claim hai. Reveal par verdict + reason.

Dam ki wall ko neeche upar se zyada mota banana chahiye.

True. Pressure depth ke saath linearly badhta hai (

P = P_{0} + ρ g h

), isliye paani base ke paas sabse zyada push karta hai; wall ko wahan sabse zyada material chahiye jahan load sabse bada ho.

Do containers of water jinka depth same hai lekin widths bahut alag hain, unka bottom pressure same hoga.

True. Bottom pressure sirf depth

h

par depend karta hai, na ki kitna paani column ke baaju mein khada hai — yahi classic hydrostatic paradox hai.

Depth double karne par fluid ka gauge pressure double ho jaata hai.

Gauge pressure ke liye True (

ρ g h

h

ke proportional hai), lekin absolute pressure ke liye False

P_{0} + ρ g h

, kyunki

P_{0}

double nahi hota.

Agar paani ki seedhi tube tilt karo, toh uske sabse neeche wale end ka pressure tube ki length par depend karta hai.

False. Ye surface se vertical drop par depend karta hai, tilted length par nahi; same vertical depth wali lambi tilted tube same pressure degi.

Fluid mein kisi point par pressure sirf neeche ki taraf act karta hai.

False. Rest mein fluid mein pressure ek point par sabhi directions mein equally act karta hai; yahi wajah hai ki hamare imaginary column par sideways forces cancel ho jaate hain aur isliye ek diver har taraf se squeeze feel karta hai.

Paani par floating oil, dono depth

h

par, same bottom pressure denge jaise pure water of depth

h

deta.

False. Oil kam dense hai, isliye same depth ka oil kam

ρ g h

contribute karta hai; fluid ki density matter karti hai, sirf height nahi.

Atmospheric pressure kuch add nahi karta us pressure mein jo ek machli 5 m neeche feel karti hai.

False. Atmosphere abhi bhi paani ki surface par press karta hai aur woh push neeche transmit hoti hai (Pascal), isliye machli

P_{0} + ρ g h

feel karti hai; sirf gauge readings

P_{0}

ko hide karti hain.

Ek open lake ki free surface par gauge pressure zero hoti hai.

True.

h = 0

par upar ka fluid column khatam ho jaata hai, isliye

ρ g h = 0

; wahan absolute pressure sirf

P_{0}

hai.

Spot the error

Har item mein ek flawed argument hai. Reveal mein specifically woh tuta hua step bataya gaya hai.

"Pool mein 50,000 L hai aur bucket mein 5 L, isliye pool ka floor bahut zyada pressure feel karta hai."

Error yeh hai ki total weight aur pressure ko confuse kiya ja raha hai. Ek point par sirf uske seedha upar wala column press karta hai; equal depth par equal pressure hoti hai, total volume chahe kuch bhi ho.

h

woh distance hai jahan main khada hoon se tank ke edge tak, lagbhag 2 m."

h

free surface se neeche vertical depth hai, seedha neeche measure ki gayi — kabhi bhi wall tak horizontal ya diagonal distance nahi.

"Question absolute pressure pooch raha hai, toh answer

ρ g h

hai."

ρ g h

gauge pressure hai. Absolute pressure ke liye

P_{0}

add karna hoga:

P = P_{0} + ρ g h

"Paani

1 g/cm^{3}

hai, isliye main

P = ρ g h

mein

ρ = 1

plug kar leta hoon."

Units mismatch — SI mein

ρ = 1000 kg/m^{3}

chahiye.

1

use karne par pressure 1000 guna kam estimate hogi.

"Derivation mein mujhe column par tank walls ki force include karni chahiye."

Walls ki forces horizontal hain, isliye woh vertical balance mein kuch contribute nahi karti aur correctly drop out ho jaati hain. Unhe vertical forces ki tarah include karna galti hai.

"Same depth ka wider column zyada weight rakhta hai, isliye bottom pressure badhni chahiye."

Wider column ka weight zyada hai lekin area bhi zyada hai (

W = ρ A h g

P = W / A

);

A

cancel ho jaata hai, aur

ρ g h

width se independent rehta hai.

"Kyunki barometer tube patli hai, mercury thodi hi chadhti hai; bada pressure padhne ke liye moti tube use karo."

Barometer height depth balance (

P_{0} = ρ g h

) par depend karti hai, tube cross-section par nahi — moti tube bhi same 760 mm degi. Area yahan bhi cancel ho jaata hai.

Why questions

Derivation mein fluid column par horizontal forces kyun cancel ho jaati hain?

Symmetry ki wajah se, curved wall par har sideways push column ke across equal aur opposite push se match hoti hai, aur unme se koi bhi vertically point nahi karti — isliye woh up–down balance mein koi role nahi nibhati.

Pressure depth par kyun depend karta hai lekin container ki shape par nahi?

Kisi point ke seedha upar wala fluid hi uski pressure mein contribute karta hai; wide ya oddly shaped vessel mein extra paani base aur walls se support hota hai, us single point ke upar stack nahi hota.

Pressure "mm of Hg" mein kyun quote kiya jaata hai?

Kyunki mercury barometer pressure ko ek column height mein convert karta hai

P = ρ g h

ke through; height batana (jab

ρ_{H g}

aur

g

known ho) pressure fully specify kar deta hai — dekho Atmospheric Pressure & Barometer.

Ek submarine ko net upward buoyant force kyun milti hai jab pressure use har taraf squeeze karta hai?

Gehre bottom face par pressure shallower top face se zyada hota hai (kyunki

P

depth ke saath badhta hai); woh pressure difference hi buoyant force hai — dekho Buoyancy & Archimedes' Principle.

Is derivation mein hum Newton's second law ki jagah first law (net force = 0) kyun use kar sakte hain?

Fluid hydrostatic hai — rest mein zero acceleration ke saath — isliye kisi bhi parcel par net force zero hai; yahi force balance

P = P_{0} + ρ g h

deta hai.

Manometer pressure ko do liquid surfaces ke beech height difference ke roop mein kyun padhta hai?

Kyunki har side

P = ρ g h

follow karti hai; equal pressures equal

ρ g h

demand karti hain, isliye pressure difference height difference

Δ h

ke roop mein dikhti hai — dekho Manometers.

Edge cases

Free surface par (

h = 0

) pressure exactly kya hai?

Gauge pressure

0

hai aur absolute pressure

P_{0}

ke barabar hai; formula

ρ g h

wahan correctly zero fluid contribution deta hai.

"Zero gravity" (freely falling ya orbiting tank) mein

ρ g h

ka kya hoga?

Effective

g = 0

hone par hydrostatic pressure difference khatam ho jaata hai — fluid har jagah equally press karta hai aur koi depth dependence nahi rehti, kyunki usse cause karne wala weight hi gone hai.

Bahut gehre ocean mein, kya constant

ρ

ke saath

P = ρ g h

exactly correct hai?

Sirf approximately — paani thoda compressible hota hai, isliye

ρ

depth ke saath badhta hai aur actual pressure constant-density formula se thoda tezi se badhta hai. Zyaatar problems ke liye constant-

ρ

answer theek hai.

Agar do immiscible liquids stack ho jaayein (oil over water), toh bottom pressure kaise nikaalte hain?

Har layer ka contribution add karo:

P = P_{0} + ρ_{oil} g h_{oil} + ρ_{water} g h_{water}

; har layer apni density aur apni vertical thickness use karti hai.

Mercury barometer mein vacuum ke top par pressure kya hai?

Essentially zero — mercury ke upar press karne ke liye (lagbhag) kuch hai hi nahi, yahi wajah hai ki atmosphere ka

P_{0}

neeche ek pura

ρ g h

column balance kar sakta hai.

Static fluid mein, kya same depth par horizontally move karne se pressure badlti hai?

Nahi — connected fluid mein equal depth par pressure har jagah same hoti hai; sirf vertical movement

h

aur isliye

P

ko change karti hai.

Recall Yahan har trap ki ek-line summary

Pressure sirf vertical depth aur density ki parwah karta hai ( $P_{0} + ρ g h$ ); woh sabhi directions mein act karta hai, container shape aur total volume ignore karta hai, "gauge" readings mein $P_{0}$ hide karta hai, aur jab $h$ ya $g$ zero ho jaaye toh khatam ho jaata hai.

Connections

Parent: the full derivation
Pascal's Law — kyun $P_{0}$ depth par feel hota hai
Buoyancy & Archimedes' Principle — depth ke saath pressure difference
Manometers — pressure difference as height difference
Atmospheric Pressure & Barometer — mm-of-Hg reasoning
Density and Specific Gravity — woh $ρ$ jo SI units mein hona chahiye