Vertical circular motion — minimum speed conditions: Foundations

1.2.18 · D1 · Physics › Newton's Laws & Dynamics › Vertical circular motion — minimum speed conditions

Yeh page woh har letter, arrow, aur idea build karta hai jo parent note use karta hai, bilkul zero se shuru karke. Agar tumne kabhi $\frac{m v ^{2}}{r}$ , $g r$ , ya "component" word bhi nahi dekha, toh yahan se shuru karo aur upar se neeche padhte jao.

1. Circle, uska center, aur radius $r$

Kisi bhi physics se pehle, hume woh shape chahiye jis par object move karta hai.

Ek ball on a string socho. Tera haath center hai. Tani hui string radius $r$ hai. Ball rim trace karti hai.

Figure — Vertical circular motion — minimum speed conditions

Figure dekho: amber dot center hai, cyan line $r$ hai, aur white circle path hai. Is poore topic ke liye sabse important habit:

2. Mass $m$ , gravity $g$ , aur weight $m g$

3. Direction, components, aur hum ek arrow ko kaise split karte hain

Kyunki gravity ka role loop ke around badalta rehta hai, hume ek clean tarika chahiye yeh poochne ka "is arrow ka kitna hissa center ki taraf point karta hai?" Woh sawaal ek component se answer hota hai.

Figure mein, object ek general angle $θ$ par baitha hai (top se measure kiya gaya). White weight arrow $m g$ do cyan shadows mein split hai:

ek radial part $m g cos θ$ — inward line par shadow,
ek tangential part $m g sin θ$ — circle ke along shadow.

4. Speed $v$ , acceleration, aur turning se $a = \frac{v ^{2}}{r}$ kyun milta hai

Ab hum inward acceleration assert nahi karenge, derive karenge.

Formula Inward acceleration

$v^{2} / r$ kyun hai Figure dekho. Thodi si time $Δ t$ mein object ek tiny arc move karta hai aur velocity arrow (length $v$ ) ek small angle $Δ θ$ se rotate hota hai.

Arc travelled circle ke along: $Δ s = v Δ t$ . Lekin ek arc bhi equals radius × angle, toh $Δ s = r Δ θ$ . Inhe equal set karo: $v Δ t = r Δ θ$ , isliye $Δ θ = \frac{v Δ t}{r}$ .
Velocity mein change. Dono velocity arrows (pehle aur baad) dono ki length $v$ hai aur same angle $Δ θ$ se differ karte hain (woh exactly position ki tarah rotate karte hain). Ek small angle ke liye, connecting arrow ki length $Δ v = v Δ θ$ hai, aur woh center ki taraf point karta hai (velocity triangle mein amber arrow dekho).
Acceleration = time per velocity mein change: $a = \frac{Δ v}{Δ t} = \frac{v Δ θ}{Δ t} = \frac{v}{Δ t} \cdot \frac{v Δ t}{r} = \frac{v ^{2}}{r} .$ Isliye $v$ squared appear karta hai: ek baar "kitna tezī se muda" se aur ek baar "har velocity kitni badi hai" se. Aur yeh inward point karta hai, toh exactly centripetal direction hai.

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

Ab inward acceleration ko mass se multiply karo, $F = ma$ use karke (§8 mein build kiya gaya).

Figure mein, object top par hai. Cyan arrow $\frac{m v ^{2}}{r}$ label kiya gaya hai — yeh required inward pull hai — object se center ki taraf (yahan neeche) point karta hai. Yeh dashed draw kiya gaya hai yaad dilane ke liye: yeh ek demand hai, koi physical arrow nahi jo tum add karo.

6. Tension $T$ aur normal force $N$ — woh pulls jo sirf ek taraf hi pull/push kar sakte hain

String (ya track) woh baki inward force supply karta hai jo gravity cover nahi karti.

Yeh figure object ko top par dono real arrows ke saath dikhata hai: white $m g$ (neeche) aur cyan $T$ (neeche, center ki taraf). Kyunki dono seedha center ki taraf point karte hain, unke radial components sirf $m g$ aur $T$ hain, aur woh add ho jaate hain. Top of the loop par equation isliye hai $T + m g = \frac{m v ^{2}}{r} .$ $T = 0$ set karo (string slack hone ki verge par) aur famous $v_{top,min} = g r$ milta hai.

7. Square-root — yeh kya undo karta hai

Answers $g r$ aur $5 g r$ ke roop mein aate hain. Make sure yeh symbol koi mystery nahi hai.

8. Newton's Second Law us form mein jo hum actually use karte hain

Yahi machine hai jo "diagram par arrows" ko equation mein convert karti hai. Loop ke har point par hum (1) real forces draw karte hain, (2) unke inward components add karte hain (§3), (3) sum ko $\frac{m v ^{2}}{r}$ ke equal set karte hain.

9. Energy: kinetic $\frac{1}{2} m v^{2}$ , potential $m g h$ , aur bottom-speed derivation

Bottom-speed result $5 g r$ ke liye ek aur tool chahiye: energy bookkeeping.

Prerequisite map — ise build order ki tarah padho

Neeche diya diagram sirf decoration nahi hai: arrows follow karo aur tumhe is page ka build order milega. Geometry aur forces (top row) centripetal requirement mein combine hote hain; Newton's law use equation mein top equation mein turn karta hai; $T = 0$ edge top speed deta hai; aur energy woh speed neeche bottom answer tak le jaati hai. Agar koi bhi box unclear hai, aage badhne se pehle uske section par wapas jaao.

Equipment checklist

Khud test karo — parent note tackle karne se pehle har cheez ka jawab de paana chahiye.

r

kya measure karta hai, aur kya yeh circle par badalta hai?

Center se object tak ki fixed distance; yeh constant rehti hai.

m g

hamesha kaunsi direction mein point karta hai?

Seedha neeche, loop ke har point par.

Ek force ka "component" kya hai?

Force ka kitna hissa ek chosen direction ke along act karta hai — woh shadow jo arrow us line par daalta hai.

Radial aur tangential directions kya hain?

Radial = center line ke along (inward); tangential = circle ke along, radial ke perpendicular.

Top se angle

θ

par gravity ka radial component kya hai?

m g cos θ

(top par full

m g

, sides par zero, bottom par

- m g

Constant speed par bhi turning ek form of acceleration kyun hai?

Velocity mein direction hoti hai; direction change karna velocity change karta hai, jo acceleration hai.

v^{2} / r

mein

v^{2}

kahan se aata hai?

v

kitni tezi se mude se, ek

v

redirect hone wali velocity ki size se.

Kya

\frac{m v ^{2}}{r}

ek real force hai jo tum draw karte ho?

Nahi — yeh woh required inward total hai jis mein real forces sum honi chahiye.

Ek string kaunsi sign restriction maanta hai?

T \geq 0

— string pull kar sakti hai lekin kabhi push nahi.

General-angle inward equation kya hai?

T + m g cos θ = \frac{m v ^{2}}{r}

Top par minimum speed kya define karta hai?

Woh moment jab

T = 0

, jahan gravity akele

\frac{m v ^{2}}{r}

supply karta hai.

Energy conservation se

5 g r

kaise nikalta hai?

v_{bot}^{2} = v_{top}^{2} + 4 g r = g r + 4 g r = 5 g r

, phir square-root.

String laga hone par hum energy conservation kyun use kar sakte hain?

Tension motion ke perpendicular hai, isliye woh zero work karta hai.

Connections

Parent topic
Centripetal force and acceleration
Newton's Second Law — net force form
Conservation of mechanical energy
Tension and constraint forces
Normal force in circular tracks
Banked curves and horizontal circular motion
Free-body diagrams