Buoyancy — Archimedes' principle, derivation from pressure difference: Question bank

2.2.7 · D5 · Physics › Fluid Mechanics › Buoyancy — Archimedes' principle, derivation from pressure d

Shuru karne se pehle, teen words jo hum baar baar use karte hain:

Displaced volume $V_{disp}$ ::: Vo volume jitna fluid object raste se hat'a deta hai — yeh object ke submerged volume ke barabar hota hai, hamesha total volume ke nahi.
Apparent weight ::: Scale kya read karta hai jab object fluid mein latka ho: $W_{app} = W_{air} - F_{B}$ . Dekho Apparent weight and weighing methods.
Average density ::: Total mass ÷ total bahari volume (koi bhi enclosed air milake). Yahi decide karta hai float hoga ya nahi, material ki apni density nahi.

True or false — justify

A block twice as deep in the same water feels twice the buoyant force.

False — buoyancy pressure difference

p_{bottom} - p_{top} = ρ_{f} g H

se aati hai, jo block ki height/volume par depend karti hai, absolute depth par nahi. Dekho Pressure in fluids — hydrostatic pressure.

Two objects of equal volume, fully submerged in the same fluid, feel equal buoyant force even if one is lead and one is wood.

True —

F_{B} = ρ_{f} V_{disp} g

mein sirf fluid density aur displaced volume hai; object ki apni density kabhi is mein nahi aati.

A floating object always displaces its own full volume of fluid.

False — floating object utna hi fluid displace karta hai jitna uska weight match kare; sirf uska submerged fraction

f = ρ_{o} / ρ_{f}

hi surface ke neeche hota hai.

The buoyant force on a fully submerged object equals its weight.

False — yeh equality sirf floating equilibrium mein hoti hai; fully submerged sinking object mein

F_{B} < W

hota hai, aur submerged daba ke rakha cork mein

F_{B} > W

hota hai.

Atmospheric pressure adds a downward push that reduces buoyancy.

False — atmosphere (

p_{0}

) top aur bottom dono faces par equally press karta hai, isliye

p_{bottom} - p_{top}

mein cancel ho jaata hai aur

F_{B}

par kabhi effect nahi dalta.

A heavier object always sinks faster/more surely than a lighter one.

False — sinking density se decide hoti hai, weight se nahi. 300-tonne steel ship float karti hai jabki 1-gram steel ball sink hoti hai, kyunki ship ki average density kam hai.

If you lower the water level so the object is only half submerged, its buoyant force halves.

True — displaced volume aadha ho jaata hai, toh

F_{B} = ρ_{f} V_{disp} g

bhi aadha ho jaata hai; yahi woh mechanism hai jisse floating body apni equilibrium depth dhundh'ti hai.

An air bubble rising in water experiences an upward buoyant force.

True — water bubble ke air se kaafi zyada dense hai, isliye

F_{B} = ρ_{water} V g

bubble ke chhote se weight se bahut zyada hoti hai; toh yeh upar uthti hai.

Spot the error

F_{B} = ρ_{o} V g

where

ρ_{o}

is the object's density."

Wrong density — buoyancy fluid ki push hai, isliye

ρ_{f}

hona chahiye. Object density sirf float vs sink decide karti hai, upthrust ki size kabhi nahi.

"A steel ship floats because steel is lighter than water."

Steel paani se dense hai; ship isliye float karti hai kyunki uski hull mein air band hai, jisse uski average density paani se kam ho jaati hai aur woh itna paani displace karti hai ki apna weight balance kar sake.

"The submerged fraction of a floating body is

ρ_{f} / ρ_{o}

Ulta hai — yeh

f = ρ_{o} / ρ_{f}

hai. Jab object fluid se kam dense ho, toh

f < 1

aana chahiye; galat version mein fraction 1 se bada ho jaata.

"Since deeper fluid has higher pressure, buoyancy grows without limit as you sink."

Incompressible fluid mein object ke across difference

ρ_{f} g H

depth se nahi badalti, isliye buoyancy constant rehti hai. (Sirf tab badlegi jab fluid compress ho, jaise deep air mein.)

"The buoyant force acts at the object's centre of mass."

Yeh centre of buoyancy par act karta hai = displaced fluid volume ka centroid, jo object ke centre of mass se tab hi milta hai jab object uniform ho. Yahi distinction stability drive karti hai — dekho Floating bodies and stability — metacentre.

"A ball hanging on a string in water: the string tension equals the ball's full weight."

Tension

W - F_{B}

(apparent weight) ke barabar hoti hai, kyunki paani ka upthrust already weight ka kuch part support kar raha hota hai.

"In the derivation the side faces contribute to the upward buoyant force."

Side faces opposite-facing pairs mein aate hain same depth par, isliye unki horizontal pressure forces cancel ho jaati hain; sirf top aur bottom faces net vertical force dete hain.

Why questions

Why does

p_{0}

vanish from the final buoyancy formula?

Yeh

p_{top}

aur

p_{bottom}

dono mein identically aata hai, isliye subtract ho jaata hai; sirf depth-difference term

ρ_{f} g H

bachta hai jo

ρ_{f} V g

ban jaata hai.

Why does the box shape used in the derivation not restrict the result to boxes?

Final

F_{B} = ρ_{f} V g

mein "box" ka koi trace nahi; kisi bhi shape ko same fluid ke identical blob se replace karo (jo equilibrium mein float karta hai) — isse dikhta hai ki surrounding fluid ko exactly usi fluid ke weight ke barabar upar push karna padta hai. Dekho Newton's laws — equilibrium of forces.

Why is buoyancy directed upward and not sideways?

Pressure sirf depth (vertically) ke saath badhta hai; horizontal pressures balance out ho jaate hain, isliye jo imbalance bachta hai woh purely vertical hai, higher pressure (deep) se lower pressure (shallow) ki taraf — yaani upar.

Why does an object feel "lighter" when weighed in water?

Paani ek upward force

F_{B}

contribute karta hai, toh scale (sirf remaining part supply karta hai)

W_{app} = W_{air} - F_{B}

read karta hai; mass change nahi hua, sirf zaroorat padne wala net supporting force kam hua.

Why does the same object float higher in seawater than in freshwater?

Seawater zyada dense hai (

ρ_{f}

bada), isliye weight balance karne ke liye chhota submerged fraction

f = ρ_{o} / ρ_{f}

kaafi hai — object ka kam hissa paani ke neeche hoga. Dekho Density and relative density.

Why does buoyancy relate to Pascal's transmission of pressure?

Dono ek resting fluid se pressure transmit karne se aate hain; yahan pressure field ka depth-dependent part net upthrust produce karta hai. Dekho Pascal's principle.

Edge cases

An object resting flat on the tank bottom with no water sealed underneath: is there buoyancy?

Koi net upthrust nahi — jab bottom face par fluid press nahi kar raha, toh higher-pressure "up" force missing hai, isliye object effectively glued down feel kar sakta hai (suction cup ki tarah) submerged hone ke bawajood.

An object of density exactly equal to the fluid's, fully submerged and released: what happens?

Yeh neutral equilibrium mein rehta hai — har jagah

F_{B} = W

hota hai, isliye naa utha naa dauba — jahan chhodo wahan hover karta rahega.

A sealed empty bottle held deep and released: does its buoyancy change as it rises?

Rigid bottle ke liye nahi — displaced volume fixed hai, isliye

F_{B}

constant rehta hai; flexible bag surface ke paas expand hogi,

V

badhega aur buoyancy bhi.

A completely weightless environment (free fall / orbit): is there buoyancy?

Nahi — jab

g \to 0

ho jaata hai, toh hydrostatic pressure gradient aur

F_{B} = ρ_{f} V g

dono vanish ho jaate hain; buoyancy ko depth-pressure difference banane ke liye gravity chahiye.

An object floating with density equal to the fluid: what fraction is submerged?

f = ρ_{o} / ρ_{f} = 1

, toh yeh fully submerged float karta hai, uska top surface fluid ke saath flush — floating aur neutral suspension ke beech ki boundary.

A dense object dropped in mercury (

ρ_{f} \approx 13600 kg/m^{3}

): why do steel and even lead float?

Unki densities (steel

\approx 7800

, lead

\approx 11300 kg/m^{3}

) mercury se kam hain, isliye

ρ_{o} < ρ_{f}

aur woh float karte hain — floating sirf density comparison par depend karta hai, object ka "heavy" feel karna matter nahi karta.

Zero-volume limit: what buoyancy on an infinitely thin sheet lying horizontally?

Jab

V_{disp} \to 0

, toh

F_{B} \to 0

— koi displaced volume nahi matlab koi upthrust nahi, consistent hai is baat se ki exploit karne ke liye koi top-to-bottom depth difference hi nahi hai.

Recall Ek-line takeaway lock karne ke liye

Buoyancy fluid ka depth-pressure difference hai jo ek force $ρ_{f} V_{disp} g$ mein sum hoti hai aur upar point karti hai; object ki apni density sirf yeh decide karne mein kaaam aati hai ki float karega ya nahi, push ki size kabhi nahi.

Connections

Parent: Buoyancy topic note
Pressure in fluids — hydrostatic pressure
Density and relative density
Apparent weight and weighing methods
Floating bodies and stability — metacentre
Pascal's principle
Newton's laws — equilibrium of forces