Foundations — Bernoulli's equation — derivation from F = ma along streamline

2.2.14 · D1 · Physics › Fluid Mechanics › Bernoulli's equation — derivation from F = ma along streamli

Parent derivation padhne se pehle, tumhe dekhna hoga ki har letter kya represent karta hai. Yeh page unhe ek-ek karke, jis order mein pehli baar aate hain, bilkul zero se build karta hai — taaki koi bhi symbol use hone se pehle uske saath ek picture ho. Agar koi term already obvious lagti ho, to uski reveal line ko neeche quick self-test ki tarah treat karo.

0 — Fluid parcel kya hota hai? (woh object jis par sab kuch hota hai)

Bernoulli mein sab kuch ek parcel par hota hai: ek imaginary chhota sa liquid ka box jo hum mentally ek alag rang se paint karte hain aur follow karte hain jab woh drift karta hai. Yeh koi real, alag cheez nahi hai — uske aas-paas ka paani bilkul wahi jaisa hai — lekin ek definite chunk choose karne se hume ek definite mass milti hai, aur mass wahi hai jo Newton ke law ko chahiye.

1 — Streamline aur $s$ (woh raasta jis par parcel chalta hai)

Ek streamline ek aisi curve hai jo har jagah fluid ki velocity ke tangent hoti hai — har point par velocity ka arrow banao, arrows ko head-to-tail jodo, aur tumhare paas streamline aa jaayegi. Parcel uske saath ride karta hai jaise car ek road par.

Hum us "road" par travel ki gayi distance ko letter $s$ se measure karte hain. To $s$ left-right ya up-down nahi hai; yeh hai "winding path par hum kitna aage gaye hain."

Topic ko iska zaroorat kyun hai. Newton ka law ek vector statement hai. Streamline ke along measure karne se hum sirf force ke us component se deal karte hain jo motion ki direction mein hai — wahi component jo parcel ki speed badal sakta hai. Isliye poori derivation one-dimensional algebra mein collapse ho jaati hai. Streamlines par build ek aur idea ke liye Equation of Continuity dekho.

2 — Chhote increments: $ds$, $dy$, $dP$, $dv$ ("$d$" matlab "ek sliver of")

Kisi quantity ke aage chhota "$d$" matlab hai "uska ek bahut chhota piece." Yeh ek aisi slice hai jitni patli ki uske across change ke alawa sab kuch effectively constant rehta hai.

• $ds$ — streamline ka ek tiny length jo parcel occupy karta hai.
• $dy$ — us length ke across height mein tiny rise.
• $dP$ — back face se front face tak pressure mein tiny change.
• $dv$ — woh tiny speed change jo parcel us slice ko cross karte waqt gain karta hai.

3 — Density $\rho$ (fluid kitna bhaari hai, per box)

Density $\rho$ (Greek letter "rho", "row" se rhyme karta hai) har unit volume mein pack ki gayi mass hai: $\rho =\text{mass}/\text{volume}$. Paani ka $\rho \approx 1000{\text{kg/m}}^3$ hota hai — ek metre side wale cube mein ek tonne aata hai.

Topic ko iska zaroorat kyun hai. Newton ke law ko mass chahiye, aur parcel ki mass $dm=\rho \times dV$ hai. Bernoulli assume karta hai ki $\rho$ constant hai (incompressible), aur yahi exactly woh cheez hai jo parent note ke Step 5 mein integral ke bahar pull karne deti hai.

4 — Area $A$, volume $dV$, mass $dm$ ("size" ko "mass" mein convert karna)

• Area $A$ — ek flat face ka size, jaise parcel-cylinder ka circular end; units ${\text{m}}^2$.
• Volume $dV$ — parcel kitni space fill karta hai. Face $A$ aur length $ds$ wale cylinder ke liye: $dV=Ads$.
• Mass $dm$ — usme kitना stuff hai: $dm=\rho dV=\rho Ads$.

Topic ko iska zaroorat kyun hai. Yeh teeno abstract "parcel" ko ek concrete number mein turn karte hain jo tum $F=ma$ mein dal sako. Area $A$ Equation of Continuity mein bhi aata hai ek throttle ki tarah jo speed set karta hai.

5 — Pressure $P$ (ek face par spread ek dhakka)

Pressure $P$ woh force hai jo ek surface par press karti hai, us surface ke area se divide ki gayi: $P=F/A$. Iske units pascals hain, $\text{Pa}={\text{N/m}}^2$. Khaas baat yeh hai ki fluid kisi bhi surface ko perpendicular push karta hai, aur yeh hamare parcel ke dono faces ko push karta hai.

Topic ko iska zaroorat kyun hai. Pressure differences poori derivation ki sirf do forces mein se ek hain. Dekho Hydrostatic Pressure ki jab fluid move nahi kar raha to $P$ kaise behave karta hai.

6 — Gravity $g$, height $y$, aur slope $\sin \theta$

• $g$ — gravity ke pull ki strength, $\approx 9.8{\text{m/s}}^2$; yeh seedha neeche point karta hai.
• $y$ — height, kisi bhi chosen floor se upar measure ki gayi. Upar wala parcel = bada $y$.
• $\theta$ — woh angle jo streamline horizontal ke saath banati hai, to $\sin \theta =dy/ds$ woh fraction of gravity hai jo road ke along hai.

Topic ko iska zaroorat kyun hai. Gravity doosri (aur aakhri) force hai. Height differences woh hai jo Torricelli ke tank ko drain karta hai — dekho Torricelli's Law.

7 — Speed $v$ aur position ke saath uska change $\frac{dv}{ds}$

$v$ parcel ki speed hai streamline ke along. Steady flow mein kisi bhi fixed spot par speed kabhi nahi badalti — lekin alag spots ki alag speeds hoti hain, to ek chalti hui parcel thodi der baad khud ko ek faster region mein paati hai.

8 — Chain rule aur derivative $\frac{dv}{dt}$ (time ko distance mein kyun convert kar sakte hain)

Acceleration hai "speed ka change per unit time", $a=\frac{dv}{dt}$. Lekin haari forces naturally distance $s$ ke terms mein likhi jaati hain, time nahi. Chain rule — do rates of change stack karne ka ek rule — swap karne deti hai: $a=\frac{dv}{dt}=\underset{\text{speed change per metre}}{\underbrace{\frac{dv}{ds}}}\times \underset{=v}{\underbrace{\frac{ds}{dt}}}=v\frac{dv}{ds}$

Yeh tool kyun aur koi doosra kyun nahi? Hume distance-based acceleration chahiye kyunki haari forces (pressure, gravity) distance $s$ ke along laid out hain. Chain rule hi ek cleanest tarika hai time-derivative ko convert karne ka jo Newton demand karta hai, us distance-derivative mein jo haari geometry supply karti hai. Yeh ek hi move final equation mein $\frac{1}{2}\rho v^2$ term produce karta hai. Newton's Second Law par build kiya gaya.

9 — Newton ka law $F=ma$ aur work–energy view

• $F=ma$ — net force mass times acceleration ke barabar hai. Dono baar zyada push karo, dono baar zyada accelerate karo; dono baar mass, aadha acceleration.
• $F=ma$ ko ek displacement se multiply karne par woh Work-Energy Theorem ban jaata hai: net work done = change in kinetic energy. Bernoulli dono taraf se padha ja sakta hai — parent note inhe "ek hi equation ki do readings" kehta hai.

10 — Integral sign $\int$ (saare slivers ko add karna)

Lamba S, $\int$, matlab "sum up." Har sliver ne ek tiny relation contribute kiya $dP+\rho gdy+\rho vdv=0$; point 1 se point 2 tak integrate karne par poori streamline ke along har sliver ka contribution add ho jaata hai: ${\int }_1^{2}dP+\rho g{\int }_1^{2}dy+\rho {\int }_1^{2}vdv=0.$

Yeh topic ko kaise feed karte hain

Equipment checklist

Right side cover karo aur khud ko test karo. Agar koi answer surprise kare, parent note kholne se pehle upar woh section dobara padho.