Foundations — Banking of roads — derivation

1.2.17 · D1 · Physics › Newton's Laws & Dynamics › Banking of roads — derivation

Is page par assume kiya gaya hai ki tumne kuch nahi dekha. Hum har symbol ko build karte hain, uski picture draw karte hain, aur tabhi parent note use karta hai.

0. "Force" aur uska arrow kya hota hai

Koi bhi letter aane se pehle, picture ko fix karo.

Hume yeh kyun chahiye: poori derivation sirf teen arrows (weight, normal, friction) ki hai jo milke ek specific arrow (inward pull) banaate hain. Agar tum arrows ko add karte nahi socha, toh baad ka koi bhi equation kaam ka nahi hoga.

1. $m$ — mass, aur $g$ — gravity ki pull

Picture: $mg$ length ka ek arrow jo ground ki taraf point kar raha hai, car ke centre se attached hai. Yeh kabhi nahi jhukta, kabhi nahi badalta — gravity ko parwah nahi ki road banked hai ya nahi.

Topic ko yeh kyun chahiye: $mg$ free-body diagram mein teen forces mein se ek hai, aur road ka push vertically isi cheez ko balance karta hai.

2. Speed $v$, radius $r$, aur "circular motion"

Figure dekho: jab bhi speed $v$ constant rehti hai, car ki direction baar baar badalti rehti hai. Direction ka change bhi ek change of motion hi hai — isliye car accelerate kar rahi hai, aur Newton's Second Law ke hisaab se acceleration ke liye hamesha ek force chahiye.

3. Acceleration $a$ — motion ke change ki rate

Picture: velocity arrow se alag ek arrow, jo dikhata hai ki velocity kis taraf nudge ho rahi hai. Circular motion ke liye yeh nudge-arrow hamesha centre ki taraf aim karta hai — isliye pull inward hona zaroori hai.

Topic ko yeh kyun chahiye: Newton's Second Law ke hisaab se, force aur acceleration $\text{force}=m\times a$ se linked hain. $a$ jaane bina hum nahi bol sakte ki inward force kitni badi honi chahiye.

4. Centripetal requirement $\frac{m{v}^2}{r}$

Yeh topic ki sabse important quantity hai, isliye hum ise slowly build karte hain.

Formula ko ek story ki tarah padho:

• Zyada mass $m$ → ghoomana mushkil → zyada pull chahiye (upar barhta hai).
• Zyada speed $v$ → har second direction ka sharp change → bahut zyada pull (yeh $v^2$ ke saath barhta hai, yaani square ke saath).
• Bada radius $r$ → gentle, lazy curve → kam pull chahiye (neeche barhta hai).

Topic ko yeh kyun chahiye: har banking formula sirf yahi sentence hai ki "road ki forces ka sideways part $\frac{m{v}^2}{r}$ ke barabar hona chahiye." Yeh woh target hai jise arrows ko hit karna hai. Poori story Centripetal force mein hai.

5. Banking angle $\theta$ aur tilted road

Picture: road ka cross-section ek ramp hai. Outer edge utha hua hai; inner edge (circle ke centre ke paas) neechi hai, jaise kisi bowl ka andar. Yahi geometry dekho Inclined plane aur Conical pendulum mein.

6. Hamare do axes chunna

Kisi bhi arrow ko split karne se pehle, hume agree karna hoga ki cheezein kin do directions mein measure karni hain.

Yeh exact choice kyun: agar hum forces ko un axes ke saath measure karein jo acceleration se match karein, toh §9 ki bookkeeping "vertical forces cancel, horizontal forces $m{v}^2/r$ banate hain" ban jaati hai — sabse simple possible statement. Koi aur axes dono ko mix kar dete.

7. Normal force $N$ — aur yeh kyun jhukta hai

Kyunki $N$ jhukta hai, iske §6 ke do axes ke hisaab se DO useful parts hain:

• ek upward part, $N\cos \theta$, jo gravity se ladhta hai;
• ek inward part, $N\sin \theta$, jo centre ki taraf point karta hai — yahi piece turning ka kaam karta hai.

Topic ko yeh kyun chahiye: yahi tilt banking ki poori trick hai. Flat road ka $N$ seedha upar point karta hai aur turn karne mein kuch nahi kar sakta; tilted road ka $N$ apna inward slice centripetal kaam ke liye deta hai.

8. Arrow ko split karna: $\sin \theta$ aur $\cos \theta$

Hum baar baar "$N\cos \theta$" aur "$N\sin \theta$" keh rahe hain. Yahan batate hain ye exact factors kyun aate hain.

Yeh tools kyun, koi aur kyun nahi: acceleration purely horizontal (inward) hai aur zero vertical. Isliye natural kaam yeh hai ki har slanted arrow ko ek vertical piece aur ek horizontal inward piece mein todna. Sine aur cosine exactly woh machines hain jo ek length aur angle se woh do pieces padhte hain. Aur kuch yeh kaam nahi karta.

9. Friction $f$ aur uska coefficient ${\mu }_s$

Picture: ramp surface par flat ek arrow. Yeh upar slope ki taraf (jab car slow ho aur andar slip kar rahi ho) ya neeche slope ki taraf (jab car fast ho aur bahar ja rahi ho) point kar sakta hai. Uski length $0$ se ${\mu }_{s}N$ tak kuch bhi ho sakti hai.

10. Axes par rakhna — $\sum F=ma$ per direction

Yeh single rule, do baar apply karne par, derivation hi hai. Upar ki sab cheezein sirf un symbols ko earn kar rahi thi jo yeh use karta hai — including $a$, do axes, $N$ ka split, aur static ${\mu }_s$.

Prerequisite map

Equipment checklist

Connections

• Banking of roads — derivation — woh topic jise yeh foundations support karte hain.
• Centripetal force — §4 mein build hua inward requirement $m{v}^2/r$.
• Uniform Circular Motion — turning accelerating kyun hota hai (§2).
• Newton's Second Law — per-axis bookkeeping (§10).
• Friction — range-setting force (§9).
• Inclined plane — same arrow-splitting geometry.
• Conical pendulum — identical $\tan \theta =v^2/rg$ setup.